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Equipartition theorem

In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

The equipartition theorem makes quantitative predictions. Like the virial theorem, it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's heat capacity can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of a single spring. For example, it predicts that every atom in a monatomic ideal gas has an average kinetic energy of $3⁄2k_{B}T$ in thermal equilibrium, where $k_{B}$ is the Boltzmann constant and T is the (thermodynamic) temperature. More generally, equipartition can be applied to any classical system in thermal equilibrium, no matter how complicated. It can be used to derive the ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. The equipartition theorem can also be used to predict the properties of stars, even white dwarfs and neutron stars, since it holds even when relativistic effects are considered.

Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy $k_{B}T$ is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. For example, the heat capacity of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in heat capacity were among the first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required. Along with other evidence, equipartition's failure to model black-body radiation—also known as the ultraviolet catastrophe—led Max Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory.

Basic concept and simple examples

The name "equipartition" means "equal division," as derived from the Latin equi from the antecedent, æquus ("equal or even"), and partition from the noun, partitio ("division, portion"). The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, on the average, once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of an inert noble gas, in thermal equilibrium at temperature $T$, has an average translational kinetic energy of $3⁄2k_{B}T$, where $k_{B}$ is the Boltzmann constant. As a consequence, since kinetic energy is equal to 1⁄2(mass)(velocity)2, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases.

In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of $1/2k_{B}T$ and therefore contributes $1/2k_{B}$ to the system's heat capacity. This has many applications.

Translational energy and ideal gases
The (Newtonian) kinetic energy of a particle of mass $m$, velocity $v$ is given by
 * $$H_{\text{kin}} = \tfrac 1 2 m |\mathbf{v}|^2 = \tfrac{1}{2} m\left( v_x^2 + v_y^2 + v_z^2 \right),$$

where $v_{x}$, $v_{y}$ and $v_{z}$ are the Cartesian components of the velocity $v$. Here, $H$ is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem.

Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute $1/2k_{B}T$ to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is $3⁄2k_{B}T$, as in the example of noble gases above.

More generally, in a monatomic ideal gas the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the total energy of an ideal gas of $N$ particles is $3⁄2 N k_{B} T$.

It follows that the heat capacity of the gas is $3⁄2 N k_{B}$ and hence, in particular, the heat capacity of a mole of such gas particles is $3⁄2N_{A}k_{B} = 3⁄2R$, where NA is the Avogadro constant and R is the gas constant. Since R ≈ 2 cal/(mol·K), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal/(mol·K). This prediction is confirmed by experiment when compared to monatomic gases.

The mean kinetic energy also allows the root mean square speed $v_{rms}$ of the gas particles to be calculated:
 * $$v_{\text{rms}} = \sqrt{\left\langle v^2 \right\rangle} = \sqrt{\frac{3 k_\text{B} T}{m}} = \sqrt{\frac{3 R T}{M}},$$

where $M = N_{A}m$ is the mass of a mole of gas particles. This result is useful for many applications such as Graham's law of effusion, which provides a method for enriching uranium.

Rotational energy and molecular tumbling in solution
A similar example is provided by a rotating molecule with principal moments of inertia $I_{1}$, $I_{2}$ and $I_{3}$. According to classical mechanics, the rotational energy of such a molecule is given by
 * $$H_{\mathrm{rot}} = \tfrac{1}{2} ( I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2 ),$$

where $ω_{1}$, $ω_{2}$, and $ω_{3}$ are the principal components of the angular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is $3⁄2k_{B}T$. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated.

The tumbling of rigid molecules—that is, the random rotations of molecules in solution—plays a key role in the relaxations observed by nuclear magnetic resonance, particularly protein NMR and residual dipolar couplings. Rotational diffusion can also be observed by other biophysical probes such as fluorescence anisotropy, flow birefringence and dielectric spectroscopy.

Potential energy and harmonic oscillators
Equipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as a spring, which has a quadratic potential energy
 * $$H_{\text{pot}} = \tfrac 1 2 a q^2,\,$$

where the constant $a$ describes the stiffness of the spring and $q$ is the deviation from equilibrium. If such a one-dimensional system has mass $m$, then its kinetic energy $H_{kin}$ is
 * $$H_{\text{kin}} = \frac{1}{2}mv^2 = \frac{p^2}{2m},$$

where $v$ and $p = mv$ denote the velocity and momentum of the oscillator. Combining these terms yields the total energy
 * $$H = H_{\text{kin}} + H_{\text{pot}} = \frac{p^2}{2m} + \frac{1}{2} a q^2.$$

Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy

\langle H \rangle = \langle H_{\text{kin}} \rangle + \langle H_{\text{pot}} \rangle = \tfrac{1}{2} k_\text{B} T + \tfrac{1}{2} k_\text{B} T = k_\text{B} T, $$ where the angular brackets $$\left\langle \ldots \right\rangle$$ denote the average of the enclosed quantity,

This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy $k_{B}T$ and hence contributes $k_{B}$ to the system's heat capacity. This can be used to derive the formula for Johnson–Nyquist noise and the Dulong–Petit law of solid heat capacities. The latter application was particularly significant in the history of equipartition.

Specific heat capacity of solids
An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of $3N$ independent simple harmonic oscillators, where $N$ denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy $k_{B}T$, the average total energy of the solid is $3N k_{B}T$, and its heat capacity is $3N k_{B}$.

By taking $N$ to be the Avogadro constant $N_{A}$, and using the relation $R = N_{A}k_{B}$ between the gas constant $R$ and the Boltzmann constant $k_{B}$, this provides an explanation for the Dulong–Petit law of specific heat capacities of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to its atomic weight. A modern version is that the molar heat capacity of a solid is 3R ≈ 6 cal/(mol·K).

However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero. A more accurate theory, incorporating quantum effects, was developed by Albert Einstein (1907) and Peter Debye (1911).

Many other physical systems can be modeled as sets of coupled oscillators. The motions of such oscillators can be decomposed into normal modes, like the vibrational modes of a piano string or the resonances of an organ pipe. On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ergodicity, is important for the law of equipartition to hold.

Sedimentation of particles
Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom $x$ contributes only a multiple of $x^{s}$ (for a fixed real number $s$) to the energy, then in thermal equilibrium the average energy of that part is $k_{B}T/s$.

There is a simple application of this extension to the sedimentation of particles under gravity. For example, the haze sometimes seen in beer can be caused by clumps of proteins that scatter light. Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also diffuse back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine the average position of a particular clump of buoyant mass $m_{b}$. For an infinitely tall bottle of beer, the gravitational potential energy is given by
 * $$H^{\mathrm{grav}} = m_\text{b} g z $$

where $z$ is the height of the protein clump in the bottle and g is the acceleration due to gravity. Since $s = 1$, the average potential energy of a protein clump equals $k_{B}T$. Hence, a protein clump with a buoyant mass of 10 MDa (roughly the size of a virus) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by the Mason–Weaver equation.

History
The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston.

(abstract only). Published in full Reprinted (reprinted in his Papers, 3, 167, 183.)

Waterston's key paper was written and submitted in 1845 to the Royal Society. After refusing to publish his work, the Society also refused to return his manuscript and stored it among its files. The manuscript was discovered in 1891 by Lord Rayleigh, who criticized the original reviewer for failing to recognize the significance of Waterston's work. Waterston managed to publish his ideas in 1851, and therefore has priority over Maxwell for enunciating the first version of the equipartition theorem. In 1859, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. In 1876, Ludwig Boltzmann expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in a system. Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong–Petit law for the specific heat capacities of solids.

The history of the equipartition theorem is intertwined with that of specific heat capacity, both of which were studied in the 19th century. In 1819, the French physicists Pierre Louis Dulong and Alexis Thérèse Petit discovered that the specific heat capacities of solid elements at room temperature were inversely proportional to the atomic weight of the element. Their law was used for many years as a technique for measuring atomic weights. However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures; at lower temperatures, or for exceptionally hard solids such as diamond, the specific heat capacity was lower.

Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mol·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction, but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K), and fell to about 3 cal/(mol·K) at very low temperatures. Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest; since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3 cal/(mol·K) and 7 cal/(mol·K), respectively.

A third discrepancy concerned the specific heat of metals. According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute $3⁄2 N_{e}k_{B}$ to the heat capacity by the equipartition theorem, where Ne is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same.

Several explanations of equipartition's failure to account for molar heat capacities were proposed. Boltzmann defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in thermal equilibrium because of their interactions with the aether. Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how. In 1900 Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem. Albert Einstein provided that escape, by showing in 1906 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid.

Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter. Nernst's 1910 measurements of specific heats at low temperatures supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists.

General formulation of the equipartition theorem
The most general form of the equipartition theorem states that under suitable assumptions (discussed below), for a physical system with Hamiltonian energy function $3⁄2$ and degrees of freedom $x_{n}$, the following equipartition formula holds in thermal equilibrium for all indices $H$ and $m$:
 * $$\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right\rangle = \delta_{mn} k_\text{B} T.$$

Here $δ_{mn}$ is the Kronecker delta, which is equal to one if $m = n$ and is zero otherwise. The averaging brackets $$\left\langle \ldots \right\rangle$$ is assumed to be an ensemble average over phase space or, under an assumption of ergodicity, a time average of a single system.

The general equipartition theorem holds in both the microcanonical ensemble, when the total energy of the system is constant, and also in the canonical ensemble, when the system is coupled to a heat bath with which it can exchange energy. Derivations of the general formula are given later in the article.

The general formula is equivalent to the following two:
 * 1) $$\left\langle x_n \frac{\partial H}{\partial x_n} \right\rangle = k_\text{B} T \quad \text{for all } n$$
 * 2) $$\left\langle x_m \frac{\partial H}{\partial x_n} \right\rangle = 0 \quad \text{for all } m \neq n.$$

If a degree of freedom xn appears only as a quadratic term anxn2 in the Hamiltonian H, then the first of these formulae implies that

 * $$k_\text{B} T = \left\langle x_n \frac{\partial H}{\partial x_n}\right\rangle = 2\left\langle a_n x_n^2 \right\rangle,$$

which is twice the contribution that this degree of freedom makes to the average energy $$\langle H\rangle$$. Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by s, applies to energies of the form anxns.

The degrees of freedom xn are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates qk and generalized momentum coordinates pk, where pk is the conjugate momentum to qk. In this situation, formula 1 means that for all k,

\left\langle p_{k} \frac{\partial H}{\partial p_{k}} \right\rangle = \left\langle q_k \frac{\partial H}{\partial q_k} \right\rangle = k_\text{B} T. $$ Using the equations of Hamiltonian mechanics, these formulae may also be written

 * $$\left\langle p_k \frac{dq_k}{dt} \right\rangle = -\left\langle q_k \frac{dp_k}{dt} \right\rangle = k_\text{B} T.$$

Similarly, one can show using formula 2 that

\left\langle q_j \frac{\partial H}{\partial p_k} \right\rangle = \left\langle p_j \frac{\partial H}{\partial q_k} \right\rangle = 0 \quad \text{ for all } \, j,k $$ and

\left\langle q_j \frac{\partial H}{\partial q_k} \right\rangle = \left\langle p_j \frac{\partial H}{\partial p_k} \right\rangle = 0 \quad \text{ for all } \, j \neq k. $$

Relation to the virial theorem
The general equipartition theorem is an extension of the virial theorem (proposed in 1870 ), which states that

\left\langle \sum_k q_k \frac{\partial H}{\partial q_{k}} \right\rangle = \left\langle \sum_k p_k \frac{\partial H}{\partial p_{k}} \right\rangle = \left\langle \sum_k p_k \frac{dq_k}{dt} \right\rangle = -\left\langle \sum_k q_k \frac{dp_k}{dt} \right\rangle, $$ where t denotes time. Two key differences are that the virial theorem relates summed rather than individual averages to each other, and it does not connect them to the temperature T. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over phase space.

Ideal gas law
Ideal gases provide an important application of the equipartition theorem. As well as providing the formula

\begin{align} \langle H^{\mathrm{kin}} \rangle &= \frac{1}{2m} \langle p_{x}^{2} + p_{y}^{2} + p_{z}^{2} \rangle\\ &= \frac{1}{2} \left( \left\langle p_{x} \frac{\partial H^{\mathrm{kin}}}{\partial p_{x}} \right\rangle + \left\langle p_{y} \frac{\partial H^{\mathrm{kin}}}{\partial p_{y}} \right\rangle + \left\langle p_{z} \frac{\partial H^{\mathrm{kin}}}{\partial p_{z}} \right\rangle \right) = \frac{3}{2} k_\text{B} T \end{align} $$ for the average kinetic energy per particle, the equipartition theorem can be used to derive the ideal gas law from classical mechanics. If q = (qx, qy, qz) and p = (px, py, pz) denote the position vector and momentum of a particle in the gas, and F is the net force on that particle, then

\begin{align} \langle \mathbf{q} \cdot \mathbf{F} \rangle &= \left\langle q_x \frac{dp_x}{dt} \right\rangle + \left\langle q_y \frac{dp_y}{dt} \right\rangle + \left\langle q_z \frac{dp_z}{dt} \right\rangle\\ &=-\left\langle q_x \frac{\partial H}{\partial q_x} \right\rangle - \left\langle q_y \frac{\partial H}{\partial q_y} \right\rangle - \left\langle q_z \frac{\partial H}{\partial q_z} \right\rangle = -3k_\text{B} T, \end{align} $$ where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition formula. Summing over a system of N particles yields

3Nk_\text{B} T = - \left\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \right\rangle. $$

By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence

-\left\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\right\rangle = P \oint_{\text{surface}} \mathbf{q} \cdot d\mathbf{S}, $$ where $dS$ is the infinitesimal area element along the walls of the container. Since the divergence of the position vector $q$ is

\boldsymbol\nabla \cdot \mathbf{q} = \frac{\partial q_x}{\partial q_x} + \frac{\partial q_y}{\partial q_y} + \frac{\partial q_z}{\partial q_z} = 3, $$ the divergence theorem implies that
 * $$P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS} = P \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) \, dV = 3PV,$$

where $dV$ is an infinitesimal volume within the container and $n$ is the total volume of the container.

Putting these equalities together yields
 * $$3Nk_\text{B} T = -\left\langle \sum_{k=1}^N \mathbf{q}_k \cdot \mathbf{F}_k \right\rangle = 3PV,$$

which immediately implies the ideal gas law for N particles:
 * $$PV = N k_\text{B} T = nRT,$$

where $n = N/N_{A}$ is the number of moles of gas and $R = N_{A}k_{B}$ is the gas constant. Although equipartition provides a simple derivation of the ideal-gas law and the internal energy, the same results can be obtained by an alternative method using the partition function.

Diatomic gases
A diatomic gas can be modelled as two masses, $m_{1}$ and $m_{2}$, joined by a spring of stiffness $a$, which is called the rigid rotor-harmonic oscillator approximation. The classical energy of this system is
 * $$H =

\frac{\left| \mathbf{p}_1 \right|^2}{2m_1} + \frac{\left| \mathbf{p}_2 \right|^2}{2m_2} + \frac{1}{2} a q^2,$$ where $p_{1}$ and $p_{2}$ are the momenta of the two atoms, and $V$ is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute $1/2k_{B}T$ to the total average energy, and $1/2k_{B}$ to the heat capacity. Therefore, the heat capacity of a gas of N diatomic molecules is predicted to be $7N·1/2k_{B}$: the momenta $p_{1}$ and $p_{2}$ contribute three degrees of freedom each, and the extension $q$ contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be $7⁄2N_{A}k_{B} = 7⁄2R$ and, thus, the predicted molar heat capacity should be roughly 7 cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5 cal/(mol·K) and fall to 3 cal/(mol·K) at very low temperatures. This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only increase the predicted specific heat, not decrease it. This discrepancy was a key piece of evidence showing the need for a quantum theory of matter.

Extreme relativistic ideal gases
Equipartition was used above to derive the classical ideal gas law from Newtonian mechanics. However, relativistic effects become dominant in some systems, such as white dwarfs and neutron stars, and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativistic ideal gas. In such cases, the kinetic energy of a single particle is given by the formula
 * $$H_{\mathrm{kin}} \approx cp = c \sqrt{p_x^2 + p_y^2 + p_z^2}.$$

Taking the derivative of $q$ with respect to the $p_{x}$ momentum component gives the formula
 * $$p_x \frac{\partial H_{\mathrm{kin}}}{\partial p_x} = c \frac{p_x^2}{\sqrt{p_x^2 + p_y^2 + p_z^2}}$$

and similarly for the $p_{y}$ and $p_{z}$ components. Adding the three components together gives
 * $$\begin{align}

\langle H_{\mathrm{kin}} \rangle &= \left\langle c \frac{p_x^2 + p_y^2 + p_z^2}{\sqrt{p_x^2 + p_y^2 + p_z^2}} \right\rangle\\ &= \left\langle p_x \frac{\partial H^{\mathrm{kin}}}{\partial p_x} \right\rangle + \left\langle p_y \frac{\partial H^{\mathrm{kin}}}{\partial p_y} \right\rangle + \left\langle p_z \frac{\partial H^{\mathrm{kin}}}{\partial p_z} \right\rangle \\ &= 3 k_\text{B} T \end{align}$$ where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for $H$ particles, it is $3 Nk_{B}T$.

Non-ideal gases
In an ideal gas the particles are assumed to interact only through collisions. The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through conservative forces whose potential $U(r)$ depends only on the distance $N$ between the particles. This situation can be described by first restricting attention to a single gas particle, and approximating the rest of the gas by a spherically symmetric distribution. It is then customary to introduce a radial distribution function $g(r)$ such that the probability density of finding another particle at a distance $r$ from the given particle is equal to $4πr^{2}ρg(r)$, where $ρ = N/V$ is the mean density of the gas. It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is
 * $$\langle h_{\mathrm{pot}} \rangle = \int_0^\infty 4\pi r^2 \rho U(r) g(r)\, dr.$$

The total mean potential energy of the gas is therefore $$ \langle H_\text{pot} \rangle = \tfrac12 N \langle h_{\mathrm{pot}} \rangle $$, where $r$ is the number of particles in the gas, and the factor $N$ is needed because summation over all the particles counts each interaction twice. Adding kinetic and potential energies, then applying equipartition, yields the energy equation
 * $$H = \langle H_{\mathrm{kin}} \rangle + \langle H_{\mathrm{pot}} \rangle =

\frac{3}{2} Nk_\text{B}T + 2\pi N \rho \int_0^\infty r^2 U(r) g(r) \, dr.$$ A similar argument, can be used to derive the pressure equation
 * $$3Nk_\text{B}T = 3PV + 2\pi N \rho \int_0^\infty r^3 U'(r) g(r)\, dr.$$

Anharmonic oscillators
An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension $1/2$ (the generalized position which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem. Simple examples are provided by potential energy functions of the form
 * $$H_{\mathrm{pot}} = C q^{s},\,$$

where $q$ and $C$ are arbitrary real constants. In these cases, the law of equipartition predicts that

k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}}}{\partial q} \right\rangle = \langle q \cdot s C q^{s-1} \rangle = \langle s C q^{s} \rangle = s \langle H_{\mathrm{pot}} \rangle. $$ Thus, the average potential energy equals $k_{B}T/s$, not $k_{B}T/2$ as for the quadratic harmonic oscillator (where $s = 2$).

More generally, a typical energy function of a one-dimensional system has a Taylor expansion in the extension $s$:
 * $$H_{\mathrm{pot}} = \sum_{n=2}^\infty C_n q^n$$

for non-negative integers $q$. There is no $n = 1$ term, because at the equilibrium point, there is no net force and so the first derivative of the energy is zero. The $n = 0$ term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that

k_\text{B} T = \left\langle q \frac{\partial H_{\mathrm{pot}}}{\partial q} \right\rangle = \sum_{n=2}^{\infty} \langle q \cdot n C_{n} q^{n-1} \rangle = \sum_{n=2}^{\infty} n C_{n} \langle q^{n} \rangle. $$ In contrast to the other examples cited here, the equipartition formula

\langle H_{\mathrm{pot}} \rangle = \frac{1}{2} k_\text{B} T - \sum_{n=3}^{\infty} \left( \frac{n - 2}{2} \right) C_{n} \langle q^{n} \rangle $$ does not allow the average potential energy to be written in terms of known constants.

Brownian motion
The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation. According to that equation, the motion of a particle of mass $n$ with velocity $v$ is governed by Newton's second law
 * $$\frac{d\mathbf{v}}{dt} = \frac{1}{m} \mathbf{F} = -\frac{\mathbf{v}}{\tau} + \frac{1}{m} \mathbf{F}_{\mathrm{rnd}},$$

where $F_{rnd}$ is a random force representing the random collisions of the particle and the surrounding molecules, and where the time constant τ reflects the drag force that opposes the particle's motion through the solution. The drag force is often written $F_{drag} = −γv$; therefore, the time constant $τ$ equals $m/γ$.

The dot product of this equation with the position vector $r$, after averaging, yields the equation

\left\langle \mathbf{r} \cdot \frac{d\mathbf{v}}{dt} \right\rangle + \frac{1}{\tau} \langle \mathbf{r} \cdot \mathbf{v} \rangle = 0 $$ for Brownian motion (since the random force $F_{rnd}$ is uncorrelated with the position $r$). Using the mathematical identities

\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{r} \right) = \frac{d}{dt} \left( r^{2} \right) = 2 \left( \mathbf{r} \cdot \mathbf{v} \right) $$ and
 * $$\frac{d}{dt} \left( \mathbf{r} \cdot \mathbf{v} \right) = v^{2} + \mathbf{r} \cdot \frac{d\mathbf{v}}{dt},$$

the basic equation for Brownian motion can be transformed into

\frac{d^{2}}{dt^{2}} \langle r^{2} \rangle + \frac{1}{\tau} \frac{d}{dt} \langle r^{2} \rangle = 2 \langle v^{2} \rangle = \frac{6}{m} k_\text{B} T, $$ where the last equality follows from the equipartition theorem for translational kinetic energy:

\langle H_{\mathrm{kin}} \rangle = \left\langle \frac{p^2}{2m} \right\rangle = \langle \tfrac{1}{2} m v^{2} \rangle = \tfrac{3}{2} k_\text{B} T. $$ The above differential equation for $$\langle r^2\rangle$$ (with suitable initial conditions) may be solved exactly:
 * $$\langle r^{2} \rangle = \frac{6k_\text{B} T \tau^{2}}{m} \left( e^{-t/\tau} - 1 + \frac{t}{\tau} \right).$$

On small time scales, with $t ≪ τ$, the particle acts as a freely moving particle: by the Taylor series of the exponential function, the squared distance grows approximately quadratically:
 * $$\langle r^{2} \rangle \approx \frac{3k_\text{B} T}{m} t^2 = \langle v^{2} \rangle t^{2}.$$

However, on long time scales, with $t ≫ τ$, the exponential and constant terms are negligible, and the squared distance grows only linearly:
 * $$\langle r^{2} \rangle \approx \frac{6k_\text{B} T\tau}{m} t = \frac{6 k_\text{B} T t}{\gamma}.$$

This describes the diffusion of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.

Stellar physics
The equipartition theorem and the related virial theorem have long been used as a tool in astrophysics. As examples, the virial theorem may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of white dwarf stars.

The average temperature of a star can be estimated from the equipartition theorem. Since most stars are spherically symmetric, the total gravitational potential energy can be estimated by integration
 * $$H_{\mathrm{grav}} = -\int_0^R \frac{4\pi r^2 G}{r} M(r)\, \rho(r)\, dr,$$

where $M(r)$ is the mass within a radius $m$ and $ρ(r)$ is the stellar density at radius $r$; $G$ represents the gravitational constant and $R$ the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula
 * $$H_{\mathrm{grav}} = - \frac{3G M^{2}}{5R},$$

where $M$ is the star's total mass. Hence, the average potential energy of a single particle is
 * $$\langle H_{\mathrm{grav}} \rangle = \frac{H_{\mathrm{grav}}}{N} = - \frac{3G M^{2}}{5RN},$$

where $N$ is the number of particles in the star. Since most stars are composed mainly of ionized hydrogen, $r$ equals roughly $M/m_{p}$, where $m_{p}$ is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature
 * $$\left\langle r \frac{\partial H_{\mathrm{grav}}}{\partial r} \right\rangle = \langle -H_{\mathrm{grav}} \rangle = k_\text{B} T = \frac{3G M^2}{5RN}.$$

Substitution of the mass and radius of the Sun yields an estimated solar temperature of T = 14 million kelvins, very close to its core temperature of 15 million kelvins. However, the Sun is much more complex than assumed by this model—both its temperature and density vary strongly with radius—and such excellent agreement (≈7% relative error) is partly fortuitous.

Star formation
The same formulae may be applied to determining the conditions for star formation in giant molecular clouds. A local fluctuation in the density of such a cloud can lead to a runaway condition in which the cloud collapses inwards under its own gravity. Such a collapse occurs when the equipartition theorem—or, equivalently, the virial theorem—is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy
 * $$\frac{3G M^{2}}{5R} > 3 N k_\text{B} T.$$

Assuming a constant density $ρ$ for the cloud
 * $$M = \frac{4}{3} \pi R^{3} \rho$$

yields a minimum mass for stellar contraction, the Jeans mass $M_{J}$
 * $$M_\text{J}^{2} = \left( \frac{5k_\text{B}T}{G m_{p}} \right)^{3} \left( \frac{3}{4\pi \rho} \right).$$

Substituting the values typically observed in such clouds ($T = 150 K$, $ρ = 0 g/cm3$) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902.

Kinetic energies and the Maxwell–Boltzmann distribution
The original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average translational kinetic energy, $3⁄2k_{B}T$. However, this is true only for ideal gas, and the same result can be derived from the Maxwell–Boltzmann distribution. First, we choose to consider only the Maxwell–Boltzmann distribution of velocity of the z-component
 * $$f (v_z) = \sqrt{\dfrac{m}{2\pi k_\text{B}T}}\;e^{\frac{-m{v_z}^2}{2k_\text{B}T}}$$

with this equation, we can calculate the mean square velocity of the $N$-component
 * $$\langle {v_z}^2 \rangle = \int_{-\infty}^{\infty} f (v_z){v_z}^2 dv_z = \dfrac{k_\text{B}T}{m}$$

Since different components of velocity are independent of each other, the average translational kinetic energy is given by
 * $$\langle E_k \rangle = \dfrac 3 2 m \langle {v_z}^2 \rangle = \dfrac 3 2 k_\text{B}T$$

Notice, the Maxwell–Boltzmann distribution should not be confused with the Boltzmann distribution, which the former can be derived from the latter by assuming the energy of a particle is equal to its translational kinetic energy.

As stated by the equipartition theorem. The same result can also be obtained by averaging the particle energy using the probability of finding the particle in certain quantum energy state.

Quadratic energies and the partition function
More generally, the equipartition theorem states that any degree of freedom $z$ which appears in the total energy $x$ only as a simple quadratic term $Ax^{2}$, where $H$ is a constant, has an average energy of $1⁄2k_{B}T$ in thermal equilibrium. In this case the equipartition theorem may be derived from the partition function $Z(β)$, where $β = 1/(k_{B}T)$ is the canonical inverse temperature. Integration over the variable $A$ yields a factor
 * $$Z_{x} = \int_{-\infty}^{\infty} dx \ e^{-\beta A x^{2}} = \sqrt{\frac{\pi}{\beta A}},$$

in the formula for $Z$. The mean energy associated with this factor is given by
 * $$\langle H_{x} \rangle = - \frac{\partial \log Z_{x}}{\partial \beta} = \frac{1}{2\beta} = \frac{1}{2} k_\text{B} T$$

as stated by the equipartition theorem.

General proofs
General derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble and for the canonical ensemble. They involve taking averages over the phase space of the system, which is a symplectic manifold.

To explain these derivations, the following notation is introduced. First, the phase space is described in terms of generalized position coordinates $q_{j}$ together with their conjugate momenta $p_{j}$. The quantities $q_{j}$ completely describe the configuration of the system, while the quantities $(q_{j},p_{j})$ together completely describe its state.

Secondly, the infinitesimal volume
 * $$d\Gamma = \prod_i dq_i \, dp_i \,$$

of the phase space is introduced and used to define the volume $Σ(E, ΔE)$ of the portion of phase space where the energy $x$ of the system lies between two limits, $H$ and $E + ΔE$:
 * $$\Sigma (E, \Delta E) = \int_{H \in \left[E, E+\Delta E \right]} d\Gamma .$$

In this expression, $ΔE$ is assumed to be very small, $ΔE ≪ E$. Similarly, $Ω(E)$ is defined to be the total volume of phase space where the energy is less than $E$:
 * $$\Omega (E) = \int_{H < E} d\Gamma.$$

Since $ΔE$ is very small, the following integrations are equivalent
 * $$\int_{H \in \left[ E, E+\Delta E \right]} \ldots d\Gamma = \Delta E \frac{\partial}{\partial E} \int_{H < E} \ldots d\Gamma,$$

where the ellipses represent the integrand. From this, it follows that $Σ$ is proportional to $ΔE$
 * $$\Sigma(E, \Delta E) = \Delta E \ \frac{\partial \Omega}{\partial E} = \Delta E \ \rho(E),$$

where $ρ(E)$ is the density of states. By the usual definitions of statistical mechanics, the entropy $E$ equals $k_{B} log Ω(E)$, and the temperature $S$ is defined by
 * $$\frac{1}{T} = \frac{\partial S}{\partial E} = k_\text{B} \frac{\partial \log \Omega}{\partial E} = k_\text{B} \frac{1}{\Omega}\,\frac{\partial \Omega}{\partial E} .$$

The canonical ensemble
In the canonical ensemble, the system is in thermal equilibrium with an infinite heat bath at temperature $T$ (in kelvins). The probability of each state in phase space is given by its Boltzmann factor times a normalization factor $$\mathcal{N}$$, which is chosen so that the probabilities sum to one
 * $$\mathcal{N} \int e^{-\beta H(p, q)} d\Gamma = 1,$$

where $β = 1/(k_{B}T)$. Using Integration by parts for a phase-space variable $x_{k}$ the above can be written as

\mathcal{N} \int e^{-\beta H(p, q)} d\Gamma = \mathcal{N} \int d[x_k e^{-\beta H(p, q)}] d\Gamma_k - \mathcal{N} \int x_k \frac{\partial e^{-\beta H(p, q)}}{\partial x_k} d\Gamma, $$ where $dΓ_{k} = dΓ/dx_{k}$, i.e., the first integration is not carried out over $x_{k}$. Performing the first integral between two limits $T$ and $a$ and simplifying the second integral yields the equation

\mathcal{N} \int \left[ e^{-\beta H(p, q)} x_{k} \right]_{x_{k}=a}^{x_{k}=b} d\Gamma_{k}+ \mathcal{N} \int e^{-\beta H(p, q)} x_{k} \beta \frac{\partial H}{\partial x_{k}} d\Gamma = 1, $$

The first term is usually zero, either because $x_{k}$ is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ensemble follows immediately

\mathcal{N} \int e^{-\beta H(p, q)} x_k \frac{\partial H}{\partial x_{k}} \,d\Gamma = \left\langle x_k \frac{\partial H}{\partial x_k} \right\rangle = \frac{1}{\beta} = k_\text{B} T. $$

Here, the averaging symbolized by $$\langle \ldots \rangle$$ is the ensemble average taken over the canonical ensemble.

The microcanonical ensemble
In the microcanonical ensemble, the system is isolated from the rest of the world, or at least very weakly coupled to it. Hence, its total energy is effectively constant; to be definite, we say that the total energy $b$ is confined between $H$ and $E+dE$. For a given energy $E$ and spread $dE$, there is a region of phase space $Σ$ in which the system has that energy, and the probability of each state in that region of phase space is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variables $x_{m}$ (which could be either $q_{k}$ or $p_{k}$) and $x_{n}$ is given by

 * $$\begin{align}

\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right \rangle &= \frac{1}{\Sigma}  \, \int_{H \in \left[ E, E+\Delta E \right]}  x_{m} \frac{\partial H}{\partial x_{n}} \,d\Gamma\\ &=\frac{\Delta E}{\Sigma}\, \frac{\partial}{\partial E} \int_{H < E} x_{m} \frac{\partial H}{\partial x_{n}} \,d\Gamma\\ &= \frac{1}{\rho} \,\frac{\partial}{\partial E} \int_{H < E} x_{m} \frac{\partial \left( H - E \right)}{\partial x_{n}} \,d\Gamma, \end{align}$$ where the last equality follows because $E$ is a constant that does not depend on $x_{n}$. Integrating by parts yields the relation
 * $$\begin{align}

\int_{H < E} x_{m} \frac{\partial ( H - E )}{\partial x_{n}} \,d\Gamma &= \int_{H < E} \frac{\partial}{\partial x_{n}} \bigl( x_m ( H - E ) \bigr) \,d\Gamma - \int_{H < E} \delta_{mn} ( H - E ) d\Gamma \\ &= \delta_{mn} \int_{H < E} ( E - H ) \,d\Gamma, \end{align}$$ since the first term on the right hand side of the first line is zero (it can be rewritten as an integral of H − E on the hypersurface where $H = E$).

Substitution of this result into the previous equation yields

\left\langle x_m \frac{\partial H}{\partial x_{n}} \right\rangle = \delta_{mn} \frac{1}{\rho} \, \frac{\partial}{\partial E} \int_{H < E} \left( E - H \right)\,d\Gamma = \delta_{mn} \frac{1}{\rho} \, \int_{H < E} \,d\Gamma = \delta_{mn} \frac{\Omega}{\rho}. $$

Since $$ \rho = \frac{\partial \Omega}{\partial E} $$ the equipartition theorem follows:

\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right\rangle = \delta_{mn} \left(\frac{1}{\Omega} \frac{\partial \Omega}{\partial E}\right)^{-1} = \delta_{mn} \left(\frac{\partial \log \Omega} {\partial E}\right)^{-1} = \delta_{mn} k_\text{B} T. $$

Thus, we have derived the general formulation of the equipartition theorem

\left\langle x_{m} \frac{\partial H}{\partial x_{n}} \right\rangle = \delta_{mn} k_\text{B} T, $$

which was so useful in the applications described above.

Requirement of ergodicity
The law of equipartition holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally likely to be populated. Consequently, it must be possible to exchange energy among all its various forms within the system, or with an external heat bath in the canonical ensemble. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the hard-sphere system of Yakov Sinai. The requirements for isolated systems to ensure ergodicity—and, thus equipartition—have been studied, and provided motivation for the modern chaos theory of dynamical systems. A chaotic Hamiltonian system need not be ergodic, although that is usually a good assumption.

A commonly cited counter-example where energy is not shared among its various forms and where equipartition does not hold in the microcanonical ensemble is a system of coupled harmonic oscillators. If the system is isolated from the rest of the world, the energy in each normal mode is constant; energy is not transferred from one mode to another. Hence, equipartition does not hold for such a system; the amount of energy in each normal mode is fixed at its initial value. If sufficiently strong nonlinear terms are present in the energy function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, the Kolmogorov–Arnold–Moser theorem states that energy will not be exchanged unless the nonlinear perturbations are strong enough; if they are too small, the energy will remain trapped in at least some of the modes.

Another simple example is an ideal gas of a finite number of colliding particles in a round vessel. Due to the vessel's symmetry, the angular momentum of such a gas is conserved. Therefore, not all states with the same energy are populated. This results in the mean particle energy being dependent on the mass of this particle, and also on the masses of all the other particles.

Another way ergodicity can be broken is by the existence of nonlinear soliton symmetries. In 1953, Fermi, Pasta, Ulam and Tsingou conducted computer simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to the Korteweg–de Vries equation led to the development of soliton mathematics.

Failure due to quantum effects
The law of equipartition breaks down when the thermal energy $k_{B}T$ is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it is a poor approximation to assume that the energy levels form a smooth continuum, which is required in the derivations of the equipartition theorem above. Historically, the failures of the classical equipartition theorem to explain specific heats and black-body radiation were critical in showing the need for a new theory of matter and radiation, namely, quantum mechanics and quantum field theory.

To illustrate the breakdown of equipartition, consider the average energy in a single (quantum) harmonic oscillator, which was discussed above for the classical case. Neglecting the irrelevant zero-point energy term since it can be factored out of the exponential functions involved in the probability distribution, the quantum harmonic oscillator energy levels are given by $k_{B}T ≪ hν$, where $hν ≫ k_{B}T$ is the Planck constant, $E$ is the fundamental frequency of the oscillator, and $ν$ is an integer. The probability of a given energy level being populated in the canonical ensemble is given by its Boltzmann factor
 * $$P(E_n) = \frac{e^{-n\beta h\nu}}{Z},$$

where $E_{n} = nhν$ and the denominator $h$ is the partition function, here a geometric series
 * $$Z = \sum_{n=0}^{\infty} e^{-n\beta h\nu} = \frac{1}{1 - e^{-\beta h\nu}}.$$

Its average energy is given by

\langle H \rangle = \sum_{n=0}^{\infty} E_{n} P(E_{n}) = \frac{1}{Z} \sum_{n=0}^{\infty} nh\nu \ e^{-n\beta h\nu} = -\frac{1}{Z} \frac{\partial Z}{\partial \beta} = -\frac{\partial \log Z}{\partial \beta}. $$

Substituting the formula for $β = 1/k_{B}T$ gives the final result
 * $$\langle H \rangle = h\nu \frac{e^{-\beta h\nu}}{1 - e^{-\beta h\nu}}.$$

At high temperatures, when the thermal energy $Z$ is much greater than the spacing $Z$ between energy levels, the exponential argument $k_{B}T$ is much less than one and the average energy becomes $hν$, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when $βhν$, the average energy goes to zero—the higher-frequency energy levels are "frozen out" (Figure 10). As another example, the internal excited electronic states of a hydrogen atom do not contribute to its specific heat as a gas at room temperature, since the thermal energy $k_{B}T$ (roughly 0.025 eV) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10 eV).

Similar considerations apply whenever the energy level spacing is much larger than the thermal energy. This reasoning was used by Max Planck and Albert Einstein, among others, to resolve the ultraviolet catastrophe of black-body radiation. The paradox arises because there are an infinite number of independent modes of the electromagnetic field in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energy $hν ≫ k_{B}T$, there would be an infinite amount of energy in the container. However, by the reasoning above, the average energy in the higher-frequency modes goes to zero as ν goes to infinity; moreover, Planck's law of black-body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.

Other, more subtle quantum effects can lead to corrections to equipartition, such as identical particles and continuous symmetries. The effects of identical particles can be dominant at very high densities and low temperatures. For example, the valence electrons in a metal can have a mean kinetic energy of a few electronvolts, which would normally correspond to a temperature of tens of thousands of kelvins. Such a state, in which the density is high enough that the Pauli exclusion principle invalidates the classical approach, is called a degenerate fermion gas. Such gases are important for the structure of white dwarf and neutron stars. At low temperatures, a fermionic analogue of the Bose–Einstein condensate (in which a large number of identical particles occupy the lowest-energy state) can form; such superfluid electrons are responsible for superconductivity.

title: Equipartition theorem


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Rootes Group

The Rootes Group or Rootes Motors Limited was a British automobile manufacturer and, separately, a major motor distributors and dealers business. Run from London's West End, the manufacturer was based in the Midlands and the distribution and dealers business in the south of England. In the decade beginning 1928 the Rootes brothers, William and Reginald, made prosperous by their very successful distribution and servicing business, were keen to enter manufacturing for closer control of the products they were selling. One brother has been termed the power unit, the other the steering and braking system.

With the financial support of Prudential Assurance, the two brothers bought some well-known British motor manufacturers, including Hillman, Humber, Singer, Sunbeam, Talbot, Commer and Karrier, controlling them through their parent, Rootes' 60-per-cent-owned subsidiary, Humber Limited.

At its height in 1960, Rootes had manufacturing plants in the Midlands at Coventry and Birmingham, in southern England at Acton, Luton and Dunstable, and a brand-new plant in the west of Scotland at Linwood. From its offices in Devonshire House, Piccadilly, in London it controlled exports and international distribution for Rootes and other motor manufacturers and its own local distribution and service operations in London, Kent, Birmingham and Manchester. There were assembly plants in nine countries outside the UK.

Rootes Group was under-capitalised and unable to survive industrial relations problems and losses from the 1963 introduction of a new aluminium-engined small car, the Hillman Imp. By mutual agreement, from mid-1964, Rootes Motors was taken over in stages by Chrysler Corporation, which bought control from the Rootes family in 1967. By the end of 1978 the last of the various elements of Chrysler UK had been sold to Peugeot and Renault.

Dealer and distributor for other manufacturers
Rootes was founded in Hawkhurst, Kent, in 1913 by William Rootes as a car sales agency independent from his father's Hawkhurst motor business. Rootes had moved his operations to Maidstone by 1914 and there he contracted to repair aero engines. In 1917 he formed Rootes Limited to buy the Maidstone branch of his father's motor business, founded by his father in 1897, to expand his aircraft engine repair business and the manufacture of aircraft parts.

In 1919 the distribution of cars and commercial vehicles resumed and operations extended to London and other part of the country. As early as 1924 Rootes had become the largest truck and car distributor in the United Kingdom. They advertised that their showrooms in Devonshire House Piccadilly could supply new cars priced from £145 to £3,000 manufactured by Rolls-Royce, Daimler, Sunbeam, Austin, Hillman, Fiat or Clyno.

Manufacturing: Humber Limited
A particular effort was put into overseas sales and it became clear the export opportunities warranted a move into car manufacture, which was achieved in 1928 by the purchase of controlling interests in first Hillman Motor Car Company Limited followed a year later by Humber Limited and Commercial Cars Limited. Hillman and Commer were made wholly owned subsidiaries of Humber Limited and the Rootes brothers' holding eventually became 60 percent of the Humber ordinary shares. The Rootes brothers could now show their ability to manufacture handsome cars with a strong sales appeal.

Humber Cycles
There was a resurgence in domestic and export demand for pedal bicycles, and in February 1932 Raleigh acquired all the Humber cycles trade marks. Manufacture was transferred to Raleigh's Nottingham works.

Rootes Securities Limited
Rootes Limited was renamed Rootes Securities Limited in 1933. During the Depression more businesses were picked up as they came available: Karrier Motors Limited 1934, Sunbeam Motor Company Limited 1934, Clement Talbot Limited 1934 and British Light Steel Pressings Limited 1937 were all bought and made subsidiaries of Humber Limited. London's Mayfair coachbuilders and Rolls-Royce and Daimler dealers Thrupp & Maberly Limited had been bought in 1926 their royal warrant always proudly displayed.

Rootes Ltd and exports
A new Rootes Limited was incorporated in 1933 to hold the very profitable core business of the Rootes brothers: the motor distribution and servicing functions, and its extension and development of export markets. It had been the largest truck and car distributor in the United Kingdom in 1924 and generated the capital to buy manufacturer Hillman, merge Hillman with manufacturer Humber and give the Rootes brothers control of Humber and the manufacturing subsidiaries they would have Humber buy.

Overseas representation of British motor manufacturers was not limited to group members.

Ownership and control, Rootes family
'Rootes Motors Limited' was the new name assumed 16 November 1949 of holding company Rootes Securities Limited. Substantially the whole of 1917's initial capital had been provided by the two Rootes brothers. Thereafter the business's expansion was financed by retained profits supplemented where necessary, for example the purchase of Hillman, by loans from The Prudential Assurance Company Limited and the company's bankers principally Midland Bank. On 24 November 1949 shares in Rootes Motors Limited were issued to the public in exchange for £3,025,000. Rootes was now a public listed company and the new capital repaid the Prudential and Midland Bank loans. The listed shares however were preference shares. The equity capital remained in the hands of the Rootes family now with new partner Prudential who had taken up all of the offered £1,000,000 of ordinary shares. External shareholders continued to hold a large proportion of Humber Limited. The preference shares issued to the public by Humber remained listed. In addition there were external shareholdings in the Rootes Acceptances vehicle exporting business and in Automobile Products of India Limited.

At this time employees totalled 17,000. Rootes owned, on average, about 80 per cent of the capital of its subsidiaries. The manufacturing subsidiaries were held through partly owned Humber Limited. Manufacture was carried out in three factories in Coventry with more at Luton, Cricklewood and Acton. There was a wholly owned assembly plant in Australia and similar facilities owned with associates in Argentina, Republic of Ireland and India. From Devonshire House in Piccadilly the original business, the marketing subsidiary, directed operations at five branches in Kent, their North Kensington service department and Birmingham and Manchester branches together with distribution companies overseas sometimes jointly owned.

Barely twelve months after listing preference shares the Rootes brothers recognised the effect death duties would have on their holdings and their businesses and the two brothers offered one quarter of Rootes Motors' ordinary shares to current holders of Rootes preference shares. Further issues of preference shares and debenture stock followed in November 1954 and November 1959. Rootes Acceptances Limited, the export financing arm, was sold.

World War II
With the outbreak of the Second World War Rootes, like most other British car manufacturers, became involved with the production of armaments. In 1940, under the Government's shadow factory scheme, Rootes built its massive assembly plant in Ryton-on-Dunsmore, near Coventry, initially manufacturing aircraft, one of the first types being the Bristol Blenheim. Production included an RAF heavy bomber, the Handley Page Halifax. These were built at a shadow factory at Speke Airport near Liverpool and at Blythe Bridge in Staffordshire from 1941 to 1943. Rootes also manufactured military vehicles, based on the Humber and Commer.

Rootes had a rare lapse of business judgement shortly after WWII. When he visited the Volkswagen factory in Wolfsburg to evaluate it for war reparations, he opined that it – and the Beetle – had no value.

Postwar acquisitions
Tilling-Stevens Limited with its subsidiary Vulcan Motors, both old-established and well-known commercial vehicle and bus manufacturers, was bought in the second half of 1950. A new acquisition in 1956 was Singer Motors Limited. Rootes' Singers, badge-engineered Hillmans, were aimed at slightly more upmarket small car buyers.

Post-war development
Following the war, Rootes also sponsored satellite manufacturing operations around the world, notably in Australasia (Rootes Australia) and the Middle East. The best known example of the latter was the Iranian-built Paykan, based on the Hillman Hunter. In 1950 it acquired Tilling-Stevens, a truck and bus manufacturer based in Maidstone, Kent.

Rootes successfully sold a range of cars priced at a slight premium to their major home market competitors, justified on the basis that they offered a level of superiority in design and finish.

Studebaker stylist Raymond Loewy was a design consultant to Rootes; evidence of his influence is most readily seen in the 1956 Audax range of cars, which included the contemporary Hillman Minx, a model also produced under licence by Isuzu Motors of Japan as the Isuzu Hillman Minx.

Engineering innovation
Rootes introduced a novel supercharged diesel engine in 1954, based on a Sulzer Brothers concept. This was the Commer TS3 2-stroke 3-cylinder engine, with 2 opposed inward facing pistons per cylinder, which drove the crankshaft through bell cranks. The 3.25 litre engine developed 90 hp, equivalent to contemporary 4-stroke diesel engines of more than twice the capacity.

The engine was used in Commer trucks as well as an industrial engine. Production ceased in 1968 after the Chrysler takeover.

Range rationalisation
Hillman when purchased had been making large cars. They introduced a straight-eight soon after Hillman became a subsidiary, but it was withdrawn as the Depression deepened. Their 2-1/2 and 3-litre cars were re-styled in the mid-1930s and renamed Humber Snipe and their small Minx was made the mainstay "bread and butter" member of the Rootes range. Sunbeam continued its sports appeal but downsizing postwar to small to medium-sized cars. Humber made the larger luxury passenger vehicles, Snipes and variants, and luxury mid-size cars ending with the compact Sceptre. The intervening break in medium-sized Humbers was filled by the postwar Sunbeams. Commer and Karrier were the commercial vehicle brands. Commer manufactured a full range of vans, trucks, tractors and bus chassis, and some badge-engineered small vehicles from the Hillman range. Karrier represented mainly municipal and special-purpose trucks, vans and buses, though towards the end included badge-engineered models from the Commer range.

Sunbeam Tiger
During the 1960s, Sunbeam's Alpine convertible was moderately successful in the US market. Rootes considered that the Alpine's sales would be improved with a more powerful model. As a result, in 1964 they introduced the Tiger, a V8 derivative powered by a 260 cuin Ford V8 engine. Carroll Shelby was involved in the development of the Tiger prototype.

A 289 cuin model followed in 1967, but few were built as it was considered inappropriate for a Chrysler vehicle to be powered by Ford. Consideration was given to installing a Chrysler V8 in the Tiger, but their engines were larger and heavier than the Ford engines, and the rear-mounted distributor would have required an unaffordable chassis design, given the limited sales.

Hillman Imp and the move to Linwood

In 1963, Rootes introduced the Hillman Imp, a compact rear-engined saloon with an innovative all-aluminium OHC engine, based on a Coventry Climax engine design (originally used for a fire pump). It was intended to be a response from Rootes to rival BMC's popular Mini, and a new factory (called the Linwood plant) on the boundary between Paisley and Elderslie, Renfrewshire was built for its assembly. The move to Linwood was forced upon the company by the British government, which had introduced the principle of "industrial development certificates" (IDCs) to build factories in depressed areas. The Linwood workforce had no experience in motor vehicle assembly and the build quality and reliability of the cars suffered. Another problem was that the component suppliers were still based in the Midlands, and the company incurred costs transporting half-finished engine castings from Linwood to be machined at Ryton and returned to Linwood once they had been assembled. Completed Imps returned south to Ryton, resulting in a 600 mi round trip. The Imp itself was underdeveloped, and the build quality and reliability problems, coupled with buyer apathy towards the design were reflected in poor sales. After a reasonably successful start in 1963–65, the Imp did not sell well. Lost production caused by frequent strike action at Linwood and escalating warranty claims left Rootes no money to develop other models.

Competition history
During the 1950s, Rootes's promotion included a strategy of participation in major UK and European car rallies. Stirling Moss and Sheila van Damm were their top drivers, and the Sunbeam-Talbot 90's win in the 1955 Monte Carlo Rally was the most significant victory.

In 1968, Rootes entered a factory team in the London-Sydney Marathon. Driving a Hillman Hunter, Andrew Cowan gained what was regarded as a surprise victory against stiff competition from other factory teams with bigger budgets.

Business strategy
William Rootes' particular business skills were in marketing. "He was a supersalesman par excellence". Rootes brothers' manufacturing business was best known for solid, dependable, well-engineered middle-market vehicles always of attractive appearance. Well-known Rootes models include the Hillman Minx, its successor the Hillman Hunter, the Humber Super Snipe and the Sunbeam Alpine.

William Rootes built the Rootes Group using specific brands for each market niche.

Management succession
Following the death in 1964 of Lord Rootes, his son, William Geoffrey Rootes, became the second Lord Rootes and became the new chairman of Rootes Motors. On 1 May 1967 Lord Rootes appointed Gilbert Hunt, a Wolverhampton-born business executive, who at the time was managing director of Massey Ferguson in the UK, to be the new managing director of the Rootes Group. Hunt's appointment was made with the support of the Chrysler Corporation, which was building its holding and control over the business during this period.

Chrysler (1967–1978)
In June 1964 Rootes Motors announced Chrysler Corporation would take a 30 per cent interest in their ordinary capital offering current shareholders double the market price and a 50 per cent share in the non-voting preference capital for almost three times market price. The purchase would leave control in British hands. On completion Rootes family holdings would still exceed those of Chrysler Corporation. The purchase was completed in October 1964. During 1966 the holdings were increased to 45 per cent of the ordinary shares and 65 per cent of the non-voting shares and in January 1967 holdings were increased to about two-thirds of Rootes Motors capital. The new name for Rootes Motors Limited – Chrysler UK Limited took effect at midnight 30 June 1970.

It has been suggested that the demise of Rootes began with losses due to industrial relations problems at their BLSP plant in London, with knock-on problems down the supply chain. By the mid-1960s, Rootes was progressively taken over by the Chrysler Corporation of the United States, following huge losses amid the commercial failure of the troubled Imp. The company's financial year ran to 31 July, and in the year ended 31 July 1967 Rootes was able to report a pre-tax profit of just £3.8 million. It was the first reported profit since 1964 and compared with a pretax loss of £10.7 million in the year ending in 1966. Lack of funds for new model investment was a striking feature of the company's final decade.

Chrysler was also keen to take control of the struggling firm as it wished to have its own wholly independent European subsidiaries like archrivals Ford and GM. Chrysler took over Simca of France and Barreiros of Spain at the same time, merging it with Rootes to create Chrysler Europe. In 1970, the Rootes name was replaced by "Chrysler United Kingdom" (correspondingly the French arm Simca was renamed "Chrysler France" at the same time), and subsequently disappeared from the public sphere. The various Rootes marques were progressively phased out during the 1970s. Only Hillman was left by 1977, when it, too, was shelved in favour of the Chrysler name. The Commer name was also phased out in the 1970s; the van and truck models mostly assuming the Dodge nameplate by 1976.

In Iacocca—an Autobiography, former Chrysler chairman Lee Iacocca was disparaging of the Rootes operation, writing that Chrysler should never have bought it. Chrysler UK continued with a range of rear-wheel-drive family cars such as the Hillman Avenger (introduced in 1970) and Hillman Hunter (introduced in 1966), while the Imp — which by now had most of its teething problems ironed out — was largely ignored by the new management. The Imp's commercial chances were not helped by the launch of new imports in the early 1970s including the Fiat 127 and Renault 5, which not only showed that front engined, front wheel drive hatchbacks would become the industry standard format in the entry level market segment, but spearheaded an influx of popular imports.

In the late 1960s, Chrysler endeavoured to market the Rootes cars in the US. These efforts proved unsuccessful. Marketing in the US was impeded by an inability to offer cars for sale during part of 1968, as the Rootes cars could not comply with exhaust emission requirements.

In the early 1970s, with the rise of interest in sub-compact cars, Chrysler offered the Hillman Avenger in North America as the Plymouth Cricket. This attempt was aborted after only two years. At the same time, Chrysler's Dodge Division offered the Dodge Colt as its "subcompact"—sourced from Mitsubishi in Japan. The Colt proved a popular and reliable car, hastening the Cricket's demise.

However, Chrysler Fevre Argentina commenced manufacturing the Hillman Avenger-based Dodge 1500, and this car continued in production until 1990. During its last 10 years of production it was badged as a Volkswagen after that firm acquired Chrysler's Argentine business. There was also a Brazilian variant until 1980.

By the early 1970s, Chrysler Europe was working on the development of two new family car ranges which would be produced at both its British and French plants, and sold under both the Chrysler and Simca marques on the relevant markets - similar to the integration of British and German model ranges which its rivals Ford and General Motors were carrying out at the time.

The Chrysler Alpine (sold in France as the Simca 1307/1308) was introduced in 1975, being produced at the former Rootes plant at Ryton as well as the former Simca plant at Poissy, near Paris. The Chrysler Sunbeam a three-door hatchback based on the Avenger floorpan, was introduced in 1977 as successor to the Hillman Imp. Also, Chrysler UK made a significant contribution to the design of Chrysler's European range. As well as the Alpine and Sunbeam, there was the saloon derivative of the Alpine, the Talbot Solara (launched in early 1980 after Chrysler sold its European operations to Peugeot), and Chrysler/Simca Horizon. The Alpine and Horizon were both voted "European Car of the Year on their launch, and the Horizon was the basis for the US Plymouth Horizon and Dodge Omni, which were very successful for Chrysler and continued in production in various guises for more than a decade.

The Imp was finally discontinued in 1976, and the Hunter followed it three years later (although it continued to be produced in Iran). Indeed, componentry for the Iranian version of the car was a successful UK export during the 1980s.

Only the Avenger-based Chrysler Sunbeam hatchback, launched in 1977 kept the Rootes lineage alive, although the Alpine name was still in use and later Alpine and Solara special edition models were given the old Rootes model names, Minx and Rapier. The rights to the Rapier name remained with the successors of the company, and were eventually resurrected again on a few limited edition Peugeot models. There was also a special Sceptre edition of the 205, 405 and the 605 SRi models. This used a black plastic badge with the chrome effect Sceptre cursive script similar to that on the sideflashes of the '60s saloons. In the case of the Peugeot cars, the Sceptre badging was applied to the bootlid and lower aft part of the front wing.

Chrysler had spent much of the 1970s unsuccessfully trying to integrate its Rootes and Simca ranges into one, coherent whole. The traditionally engineered, rear-wheel-drive cars of the British division had limited appeal outside the UK, although the Avenger and Hunter—the first locally assembled car to reach a total of 30,000 units sold in its 12-year lifespan—were both relatively successful in New Zealand. Hunter production continued there and in Ireland until 1979, and it was built in Iran by Iran Khodro as the Peykan for many years more.

Unfortunately, with its problems in the US, Chrysler did not have the capital to invest in refreshing their entire product range, and sales of the older designs stagnated in the face of more modern competition. Also, the production facilities were outmoded, industrial relations problems were persistent, and the products had a poor reputation for quality.

In the face of massive losses, and the risk of high unemployment in the surrounding areas if the factories closed, the Ryton and Linwood factories were the subject of frequent government bail-outs.

Peugeot-Talbot (1978–2007)
Despite the government assistance, the weight of problems bearing on Chrysler Europe resulted in its collapse in 1977, leading to the company's 1978 takeover by PSA Peugeot-Citroen. PSA soon wielded the axe over the troubled Linwood factory in Scotland, and exhumed the Talbot marque from the pages of Rootes' history to re-badge the former Chrysler and Simca badged models. Whilst Ryton was saved, PSA took little interest in the heavy commercial vehicles and the former Commer/Dodge/Karrier truck and van factory was run in conjunction with the trucks division of Renault. After the withdrawal of the last Dodge-derived trucks (latterly badged as Renaults) it became an engine production plant for Renault Véhicules Industriels.

The first Rootes model to be discontinued under Peugeot's ownership was the Hunter in 1979, and its production tooling subsequently went to Iran, where the Paykan went into local production, which continued until 2004. It remains a common sight throughout the Middle East, especially as a taxi. The closure of Linwood in 1981 spelled the end (in Europe at least) for the Avenger. Chrysler had retained the rights to the car, and continued its production in Argentina. The demise of Linwood also meant the demise of the Talbot Sunbeam after just four years in production. Its successor was the French built Talbot Samba, based on the Peugeot 104, and which went on sale in the UK in February 1982. By this stage, however, Peugeot was a year away from unveiling its highly acclaimed new 205 supermini, although it was originally planned that Peugeot would launch a Talbot badged supermini on the 205 platform as well as a Citroen some time around the mid-1980s.

The Simca-based models (the Horizon, Alpine and Solara) continued to be built at Ryton, using the resurrected Talbot badge from 1979. However, the Talbot-badged models declined in popularity over the next few years, with sales dropping from nearly 120,000 on the UK market in 1979 to around 25,000 in 1984, while at the same time the Peugeot brand was gaining ground thanks to the success of the 205.

By 1985 PSA had announced that it would abandon the three-marque strategy on passenger cars, and the Horizon replacement, developed as the Talbot Arizona, became the Peugeot 309 when it was launched in the autumn of 1985, and was the first Peugeot-badged car to be assembled at the Ryton plant. The Arizona's styling was mismatched with the rest of the range, making it resemble an enlarged Simca 1108. The Talbot badge was discontinued on passenger cars, with the Alpine, Solara and Samba all finishing production in May 1986, and the last Horizon models leaving the production lines in Spain and Finland a year later. The Talbot brand survived on the Express van - part of a venture between Peugeot and Fiat - until 1994.

The Ryton plant was initially retained and went on to assemble the Peugeot 405 alongside the 309 from 1987, as well as the 309's successor (the Peugeot 306) from late 1992, and finally the Peugeot 206 from 1998.

Ryton began assembling its last Peugeot, the 206, in 1998. At the height of the car's success, the plant was working at capacity to satisfy demand. Ryton's importance in PSA's overall strategy was as an assembly operation with limited production capacity compared to PSA's main factories in France and Spain - particularly as its location between two main roads made expansion impossible.

In April 2006, after years of speculation surrounding Ryton's future, the PSA Group announced that production of the Peugeot 206 would be shifted to Slovakia and the Ryton plant would close within the next year.

Production at the plant ceased on 12 December 2006. It marked the end of nearly 60 years of car manufacturing at Ryton, and severed the motor industry's final remaining direct link with the Rootes Group. More than 2,000 jobs were lost as a result of the factory's closure. It was demolished soon after. The only physical remnant of the Rootes Group still in existence is the Whitley research and development centre - originally established during the Chrysler era, but subsequently sold by PSA and is now owned by Jaguar Land Rover.

The intellectual property owned by the former Rootes Group (including dormant marque and model names) was divided up between Chrysler and PSA Peugeot-Citroën following the divestment of the company's assets in 1978; and still appear from time to time on both companies' products - for example Peugeot has used the names "Minx", "Rapier" and "Sceptre" on various occasions, and on one occasion considered a second resurrection of the Talbot marque as a new budget brand - whilst Chrysler has reused the "Avenger" nameplate on the Dodge Avenger, and more recently on the Jeep Avenger in 2023.

In 2021 the successors of both conglomerates, namely Fiat Chrysler Automobiles, and Groupe PSA merged to form Stellantis - effectively bringing together the surviving fragments of the Rootes Group back under one umbrella.

Commemorations
Maidstone, where William Rootes set up his business in 1914, managed to keep the Rootes name on their building until 2007. Built in 1938-39 for Rootes it is listed Grade II. On 1 January 2007, in line with the other 40 dealerships within its business group, the name was changed from Rootes Maidstone, to Robins & Day Maidstone. Robins & Day is wholly owned and operated by Peugeot UK, as opposed to many car dealerships which are franchises. As of April 2020 it is being converted into luxury penthouses and will be completed in 2024.

Rootes' contribution to Coventry's history is commemorated by the University of Warwick in the naming of Rootes Hall, one of its largest halls of residence, on the main campus site on the outskirts of Coventry.

The name lives on in the short access to the Rootes Estate, a housing estate built in the early 1990s on most of the Ladbroke Grove site of the Clément-Talbot car factory. The main access to the redevelopment, Shrewsbury Street, is named after the founder of Clément-Talbot, Charles Chetwynd-Talbot, 20th Earl of Shrewsbury. Soon after the Second World War the group of Clément-Talbot factory buildings was converted to a Rootes Group London administration and service depot which was later bought by Warwick Wright. One of the original buildings remains, the Talbot administration block now known as Ladbroke Hall, with the earl's crest high above its main entrance. A tenant, Sunbeam studios, is named after the Sunbeam-Talbot car briefly produced there from 1938 to 1939 and 1945 to 1946. In the housing area there are short links one named Humber Drive and a Hillman Drive each side of a small park and there is a Sunbeam Crescent.

title: Rootes Group


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Fraser River (Colorado)

The Fraser River is a tributary of the Colorado River, approximately 32.5 mi long, in north central Colorado in the United States. It drains a large portion of the Middle Park basin in Grand County in the Rocky Mountains west of Boulder and southwest of Rocky Mountain National Park. It rises at the continental divide on the north side of Berthoud Pass in the Arapaho National Forest. It flows NNW past Winter Park, Fraser, and Tabernash, and joins the Colorado River from the south two miles west of Granby.

Fishing
The Fraser holds wild rainbows, browns, brooks, and cutthroats. The Fraser River starts near Berthoud Pass. As it runs north for its first 8 miles, it is on National Forest land and is publicly accessible. The river is narrow and shallow here, but worthy of fishing.

From Winter Park to Fraser, the river can be accessed by the Fraser River Trail (hiking or mountain bikes), from USFS campgrounds, or road turnouts. The only exception is as the river flows through private land in the town of Winter Park. You can go fishing in the forested lands, or where the river starts into the valley.

Access to the Fraser River is limited downstream from the town of Fraser. Some of the best fishing is in the canyon downstream from Tabernash, however this is on private land.

title: Fraser River (Colorado)


text:
Singularitarianism

Singularitarianism is a movement defined by the belief that a technological singularity—the creation of superintelligence—will likely happen in the medium future, and that deliberate action ought to be taken to ensure that the singularity benefits humans.

Singularitarians are distinguished from other futurists who speculate on a technological singularity by their belief that the singularity is not only possible, but desirable if guided prudently. Accordingly, they may sometimes dedicate their lives to acting in ways they believe will contribute to its rapid yet safe realization.

Time magazine describes the worldview of Singularitarians by saying "even though it sounds like science fiction, it isn't, no more than a weather forecast is science fiction. It's not a fringe idea; it's a serious hypothesis about the future of life on Earth. There's an intellectual gag reflex that kicks in anytime you try to swallow an idea that involves super-intelligent immortal cyborgs, but... while the Singularity appears to be, on the face of it, preposterous, it's an idea that rewards sober, careful evaluation".

Definition
The term "Singularitarian" was originally defined by Extropian thinker Mark Plus (Mark Potts) in 1991 to mean "one who believes the concept of a Singularity". This term has since been redefined to mean "Singularity activist" or "friend of the Singularity"; that is, one who acts so as to bring about the singularity.

Singularitarianism can also be thought of as an orientation or an outlook that prefers the enhancement of human intelligence as a specific transhumanist goal instead of focusing on specific technologies such as A.I. There are also definitions that identify a singularitarian as an activist or a friend of the concept of singularity, that is, one who acts so as to bring about a singularity. Some sources described it as a moral philosophy that advocates deliberate action to bring about and steer the development of a superintelligence that will lead to a theoretical future point that emerges during a time of accelerated change.

Inventor and futurist Ray Kurzweil, author of the 2005 book The Singularity Is Near: When Humans Transcend Biology, defines a Singularitarian as someone "who understands the Singularity and who has reflected on its implications for his or her own life" and estimates the singularity will occur around 2045.

History
An early singularitarian articulation that history is making progress toward a point of superhuman intelligence is found in Hegel's work, The Phenomenology of Spirit. In 1993, mathematician, computer scientist, and science fiction author Vernor Vinge hypothesized that the moment might come when technology will allow "creation of entities with greater than human intelligence" and used the term "the Singularity" to describe this moment. He suggested that the singularity may pose an existential risk for humanity, and that it could happen through one of four means:

 * 1) The development of computers that are "awake" and superhumanly intelligent.
 * 2) Large computer networks (and their associated users) may "wake up" as a superhumanly intelligent entity.
 * 3) Computer/human interfaces may become so intimate that users may reasonably be considered superhumanly intelligent.
 * 4) Biological science may find ways to improve upon the natural human intellect.

Singularitarianism coalesced into a coherent ideology in 2000, when artificial intelligence (AI) researcher Eliezer Yudkowsky wrote The Singularitarian Principles, in which he stated that a Singularitarian believes that the singularity is a secular, non-mystical event which is possible and beneficial to the world and is worked towards by its adherents. Yudkowsky's conceptualization of the singularity offered a broad definition developed to be inclusive of various interpretations. There are theorists such as Michael Anissimov who argued for a strict definition, one that refers only to the advocacy of the development of posthuman (greater than human) intelligence.

In June 2000 Yudkowsky, with the support of Internet entrepreneurs Brian Atkins and Sabine Atkins, founded the Machine Intelligence Research Institute to work towards the creation of self-improving Friendly AI. MIRI's writings argue for the idea that an AI with the ability to improve upon its own design (Seed AI) would rapidly lead to superintelligence. These Singularitarians believe that reaching the singularity swiftly and safely is the best possible way to minimize net existential risk.

Many people believe a technological singularity is possible without adopting Singularitarianism as a moral philosophy. Although the exact numbers are hard to quantify, Singularitarianism is a small movement, which includes transhumanist philosopher Nick Bostrom. Inventor and futurist Ray Kurzweil, who predicts that the Singularity will occur circa 2045, greatly contributed to popularizing Singularitarianism with his 2005 book The Singularity Is Near: When Humans Transcend Biology.

"What, then, is the Singularity? It's a future period during which the pace of technological change will be so rapid, its impact so deep, that human life will be irreversibly transformed. Although neither utopian or dystopian, this epoch will transform the concepts we rely on to give meaning to our lives, from our business models to the cycle of human life, including death itself. Understanding the Singularity will alter our perspective on the significance of our past and the ramifications for our future. To truly understand it inherently changes one's view of life in general and one's particular life. I regard someone who understands the Singularity and who has reflected on its implications for his or her own life as a "singularitarian.""

With the support of NASA, Google, and a broad range of technology forecasters and technocapitalists, the Singularity University opened in June 2009 at the NASA Research Park in Silicon Valley with the goal of preparing the next generation of leaders to address the challenges of accelerating change.

In July 2009, many prominent Singularitarians participated in a conference organized by the Association for the Advancement of Artificial Intelligence (AAAI) to discuss the potential impact of robots and computers and the impact of the hypothetical possibility that they could become self-sufficient and able to make their own decisions. They discussed the possibility and the extent to which computers and robots might be able to acquire any level of autonomy, and to what degree they could use such abilities to possibly pose any threat or hazard (i.e., cybernetic revolt). They noted that some machines have acquired various forms of semi-autonomy, including being able to find power sources on their own and being able to independently choose targets to attack with weapons. They warned that some computer viruses can evade elimination and have achieved "cockroach intelligence". They asserted that self-awareness as depicted in science fiction is probably unlikely, but that there were other potential hazards and pitfalls. Some experts and academics have questioned the use of robots for military combat, especially when such robots are given some degree of autonomous functions. The President of the AAAI has commissioned a study to look at this issue.

Reception
There are several objections to Kurzweil's singularitarianism, and these even include criticisms from optimists within the A.I. field. For instance, Pulitzer Prize winning author Douglas Hofstadter argued that Kurzweil's predicted achievement of human-level A.I. by 2045 is not viable. Even Gordon Moore, the namesake of Moore's Law that predicated the notion of singularity, maintained that it will never occur. According to some observers, these criticisms do not diminish enthusiasm for singularity because it has assumed a quasi-religious response to the fear of death, allowing its adherents to enjoy the benefits of religion without its ontological burdens. Science journalist John Horgan provided more insights into this notion as he likened singularitarianism to a religion: "Let's face it. The singularity is a religious rather than a scientific vision. The science-fiction writer Ken MacLeod has dubbed it "the rapture for nerds," an allusion to the end-time, when Jesus whisks the faithful to heaven and leaves us sinners behind. Such yearning for transcendence, whether spiritual or technological, is all too understandable. Both as individuals and as a species, we face deadly serious problems, including terrorism, nuclear proliferation, overpopulation, poverty, famine, environmental degradation, climate change, resource depletion, and AIDS. Engineers and scientists should be helping us face the world's problems and find solutions to them, rather than indulging in escapist, pseudoscientific fantasies like the singularity."

Kurzweil rejected this categorization, stating that his predictions about the singularity are driven by the data that increases in computational technology have been exponential in the past. He claimed that critics who challenge his view mistakenly take an intuitive linear view of technological advancement rather than accounting for that exponential growth.

title: Singularitarianism


text: User talk:142.32.208.234

title: User talk:142.32.208.234


text:
Air India Flight 855

Air India Flight 855 was a scheduled passenger flight from Bombay (now Mumbai), India, to Dubai, United Arab Emirates. On 1 January 1978, the Boeing 747 operating the flight crashed about 3 km off the coast of Bandra, less than two minutes after take-off, killing all 213 passengers and crew on board. An investigation into the crash determined the most likely probable cause was the captain becoming spatially disoriented and losing control of the aircraft after the failure of one of the flight instruments. It was the deadliest air accident both in Air India's and India's history until Flight 182 in 1985 and then Charkhi Dadri mid-air collision in 1996.

Aircraft and crew
The aircraft involved was a Boeing 747-237B, registration named Emperor Ashoka. The plane first flew on 8 March 1971 and was the first 747 delivered to Air India on 22 April 1971. The aircraft was equipped with four Pratt & Whitney JT9D-7J engines.

The flight crew consisted of the following persons:
 * The captain was 51-year old Madan Lal Kukar. He had joined Air India in 1956, and was experienced, having almost 18,000 flight hours.
 * The first officer was 43-year-old Indu Virmani, a former Indian Air Force commander who joined Air India in 1976. He had more than 4,500 flight hours.
 * The flight engineer was 53-year-old Alfredo Faria, who joined Air India in 1955 and had 11,000 flight hours, making him one of Air India's most senior flight engineers at the time of the accident.

History of the flight
The aircraft departed from Bombay's Santa Cruz Airport (later Sahar Airport, now called Chhatrapati Shivaji Maharaj International Airport). The destination was Dubai International Airport in Dubai.

Approximately one minute after takeoff from runway 27, Captain Kukar made a scheduled right turn upon crossing the Bombay coastline over the Arabian Sea, after which the aircraft briefly returned to a normal level position. Soon it began rolling to the left, and never regained level flight.

The cockpit voice recorder recovered from the wreckage revealed that Captain Kukar was the first to notice a problem, when he said, "What's happened here, my instruments ..." The captain was explaining that his attitude direction indicator (ADI) had "toppled", meaning that it was still showing the aircraft in a right bank. First Officer Virmani, whose presumably functional AI was now showing a left bank (and not noticing the captain's concern), said, "Mine has also toppled, looks fine." It is believed that the Captain mistakenly took this to mean that both primary AIs were indicating a right bank, in effect confirming what he believed he was seeing. It was after sunset and the aircraft was flying over a dark Arabian Sea, leaving the aircrew unable to visually cross-check their AI instrument readings with the actual horizon outside the cockpit windows.

The Boeing 747 had a third backup AI in the center instrument panel between the two pilots, and the transcripts of the cockpit conversation showed Flight Engineer Faria telling the captain, "Don't go by that one, don't go by that one..." trying to direct his attention towards that third AI, or perhaps to another instrument called the turn and bank indicator, just five seconds before the aircraft impacted the sea.

The captain's mistaken perception of the aircraft's attitude resulted in him using the aircraft flight control system to add more left bank and left rudder, causing the Boeing 747 to roll further left into a bank of 108 degrees and rapidly lose altitude. Just 101 seconds after leaving the runway, the jet hit the Arabian Sea at an estimated 35-degree nose-down angle. There were no survivors among the 190 passengers and 23 crew members.

Investigation
The partially recovered wreckage revealed no evidence of explosion, fire, or any electrical or mechanical failure; and an initial theory of sabotage was ruled out.

The investigation concluded that the probable cause was "due to the irrational control inputs by the captain following complete unawareness of the altitude as his ADI had malfunctioned. The crew failed to gain control based on the other flight instruments."

US Federal District Judge James M. Fitzgerald, in a 139-page decision issued 1 November 1985, rejected charges of negligence against the Boeing Company, Lear Siegler Inc, the manufacturer of the altitude   director indicator, and the Collins Radio division of Rockwell International, which manufactured the backup system in a suit related to the crash. Steven C. Marshall, the attorney for Boeing asserted that the crash had been caused by Captain Madan Kukar, who he said was "flying illegally under the influence of diabetic drugs, a condition compounded by his alcoholic intake and dieting in the 24 hours before the flight," and not due to equipment malfunctions. The suit was dismissed in 1986.

title: Air India Flight 855