Carlo Rovelli

Francesca Vidotto

Compact phase space, cosmological constant, discrete time

Keywords: Compact phase space, cosmological constant, discrete time

published or not. The documents may come

I. INTRODUCTION

The presence of the cosmological constant can affect the quantum kinematics of gravity. Here we show that it enters naturally in loop quantum gravity (LQG) by determining the size of a compact phase space and the dimension of the corresponding finite dimensional Hilbert space. This yields the discretization of the extrinsic curvature and can be related to time discreetness.

Recent results indicate that a positive cosmological constant simplifies, rather than complicating, our understanding of quantum gravity. Fairbairn and Meusburger [START_REF] Fairbairn | Quantum deformation of two four-dimensional spin foam models[END_REF] and Han [START_REF] Han | 4-dimensional Spin-foam Model with Quantum Lorentz Group[END_REF][START_REF] Han | Cosmological Constant in LQG Vertex Amplitude[END_REF][START_REF] Han | Covariant Loop Quantum Gravity, Low Energy Perturbation Theory, and Einstein Gravity[END_REF], building on [START_REF] Rovelli | The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis[END_REF][START_REF] Smolin | Linking topological quantum field theory and nonperturbative quantum gravity[END_REF] and [START_REF] Noui | Cosmological deformation of Lorentzian spin foam models[END_REF], have shown that the cosmological constant makes covariant LQG finite. Haggard, Han, Kamiński and Riello [START_REF] Haggard | SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry[END_REF] have given a straightforward construction of the LQG dynamics in the presence of the cosmological constant, related to the geometry of constant curvature triangulations, a key idea introduced by Bahr and Dittrich [START_REF] Bahr | Regge calculus from a new angle[END_REF], which grounds the present work. The LQG kinematics needs to be modified to take into account the presence of a cosmological constant; this was realised long ago by Borissov, Major and Smolin [START_REF] Borissov | The geometry of quantum spin networks[END_REF][START_REF] Major | Quantum deformation of quantum gravity[END_REF][START_REF] Major | On the q-quantum gravity loop algebra[END_REF] and the problem has been recently explored in detail by Dupius, Girelli, Livine and Bonzom [START_REF] Dupuis | Quantum hyperbolic geometry in loop quantum gravity with cosmological constant[END_REF][START_REF] Dupuis | Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries[END_REF][START_REF] Bonzom | Towards the Turaev-Viro amplitudes from a Hamiltonian constraint[END_REF][START_REF] Bonzom | Deformed phase space for 3d loop gravity and hyperbolic discrete geometries[END_REF] for negative cosmological constant.

A discretization of spacetime in terms of flat simplices is not suitable for a theory with cosmological constant because flat geometry solves the field equations only with vanishing cosmological constant. This problem can be solved choosing a discretization with simplices with constant curvature. Here we show with a positive cosmological constant, a constant curvature discretization leads to a modification of the LQG phase space. The phase space turns out to be compact for each link. The conventional LQG phase space is modified by curving the conjugate momentum space. Curved momentum space has been repeatedly considered in quantum gravity, for instance in the relative locality framework [START_REF] Amelino-Camelia | The principle of relative locality[END_REF]. Here it is not the momentum space of a particle to be curved, but rather the space of the conjugate momentum of the gravitational field itself. We study the quantization of the resulting phase space, and we write explicitly modified quantum geometrical operators. We show that these are related to a q deformation of SU (2). A q deformations has been derived in LQG as a way to implement the dynamics of the theory with cosmological constant in [START_REF] Noui | Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity[END_REF][START_REF] Pranzetti | Turaev-Viro amplitudes from 2+1 loop quantum gravity[END_REF]. Here we have shown that it is also directly implied by the constant curvature of the individual simplices, and we have given the corresponding form of the geometrical operators of the gravitational theory, in the presence of a cosmological constant.

This result has physical consequences: it is not just the intrinsic geometry, but also the extrinsic geometry to be discrete in quantum gravity (for a hint of this see [START_REF] Rovelli | Evidence for Maximal Acceleration and Singularity Resolution in Covariant Loop Quantum Gravity[END_REF]). This indicates that not just space, but also proper time, is predicted to be discrete in the theory.

We study the quantization of the geometry in 2+1 Euclidean dimensions with positive cosmological constant, that results from building on constant curvature triangulations, but the results that we obtain may be relevant for the physical 3+1 Lorentzian case.

II. CONSTANT CURVATURE GEOMETRY

We start with the geometry of a constant curvature triangle. For this, it is convenient to fix units where the constant curvature has unit value. The small-curvature limit is then the limit where the triangle is small. Consider therefore a metric sphere with unit radius R = 1. A constant curvature triangle has three vertices on this sphere. It is a portion of the sphere bounded by three arcs of maximal circles joining two vertices. The geometry of the triangle is determined, up to isometries, by giving the length L l , l = 1, 2, 3, of these three (oriented) arcs. Importantly, these lengths are bounded.

Since the radius of the sphere determines a unit of length, these lengths can be given a-dimensionally: they are determined by the three angles α l = L l /R they define at the center of the sphere. There is also an elements k l of SO(3) associated which each arc. This is simply the rotation by an angle α l around the axis normal to the plane of the great circle of the arc. The intrinsic geometry of the triangle can therefore be given by associating (non independent) SO(3) elements k l to each of its sides.

Consider a two-dimensional surface Σ immersed in a arXiv:1502.00278v2 [gr-qc] 4 Feb 2015 three dimensional Riemannian manifold. This surface inherits an intrinsic and an extrinsic geometry from the 3d manifold. Fix a 3d triangulation of the 3d manifold, inducing a 2d triangulation on Σ. The holonomy of the 3d spin connection along each side l of the triangulation is an element h l of SO(3). The index l runs over all the segments (arcs) of the triangulation. We choose to approximate the intrinsic geometry of each 3-cell with a constant curvature metric, and therefore the geometry of each triangle with a triangle of uniform curvature. The the full intrinsic and extrinsic geometry of Σ is defined by a couple of SO(3) elements, (k l , h l ), associated to each arc: h l is the holonomy of the 3d connection, k l is the rotation associated to the curved arc l.

It is customary in loop quantum gravity (LQG) to consider the trivalent graph dual to the triangulation. In two space dimension, each link l of the graph corresponds to an arc l of the triangulation. The geometrical data are therefore an element (k l , h l ), in SO(3) × SO(3) for each link l of the graph.

In the limit in which the triangles are small compared to the constant curvature scale, the group elements k l are near the identity and we can approximate them as

k l = e J l • τ ∼ 1 1 + J l = 1 + J l • τ ,
where τ is a basis in the so(3) ∼ su(2) algebra. In this limit, which we call R → ∞, the geometry of the surface is approximated by an element of the group, h l , and an element of the algebra, J l , for each arc l. These are the standard geometrical data of LQG. Let us shift from SO(3) to SU (2) as is conventionally done in LQG. We write the limit in the form

SU (2) × SU (2) → R→∞ su(2) × SU (2) = T * SU (2) (k, h) → R→∞ (J, h)
Thus, what we do here is to modify the usual LQG kinematics by replacing the algebra with the group. The role of SU (2) × SU (2) in Euclidean 3d gravity with positive cosmological constant has been pointed out by Meusburger and Schroers in [START_REF] Meusburger | Poisson structure and symmetry in the Chern?Simons formulation of (2 1)-dimensional gravity[END_REF][START_REF] Meusburger | Quaternionic and Poisson?Lie structures in three-dimensional gravity: The cosmological constant as deformation parameter[END_REF], as the local isometry group, or as the gauge group of the Chern-Simon formulation of the theory.

III. COMPACT PHASE SPACES

The key difference between the algebra and the group is that the second is compact. This has significative consequences in the quantum theory. Their importance for finiteness has been pointed out in [START_REF] Nozari | Natural Cutoffs via Compact Symplectic Manifolds[END_REF]. These are the consequences we explore here.

A compact phase space is the classical limit of a quantum system with a finite dimensional Hilbert space. This can be seen in many ways; the simplest is to notice that a compact phase space has a finite (Liouville) volume, and therefore can accommodate a finite number of Planck size cells, and therefore a finite number of orthogonal quantum states. The familiar example of quantum systems with finite dimensional Hilbert space is given by angular momentum, for systems with fixed total angular momentum, where the quantum state space is the Hilbert space H j that carries the spin-j representation of SU (2).

In standard LQG, the kinematical data are given by an element of Γ ≡ su(2) × SU (2) on each link. Γ is the phase space of the theory, for each link. Since it is a a cotangent space, it carries a natural symplectic structure. The corresponding quantization defines the quantum theory of gravity in the loop representation. This is defined on the Hilbert space L 2 [SU (2)], where the group elements act multiplicatively and the algebra elements act as left invariant vector fields. Here we want to modify this structure by replacing the algebra su(2) with the group SU (2). The problem we address is therefore to determine the phase space structure of SU (2) × SU (2) and its quantization.

As a preliminary exercise, we address this problem for the simplified case of U (1) × U (1).

IV. U(1) × U(1)
Here we define and quantize the phase space U (1) × U (1). We write elements of this space as a couple (h, k) of complex numbers with unit norms, with (h = e iα , k = e iβ ). Let us start by determining the phase space structure, namely writing the symplectic two-form. We are guided to do so by the fact that in the limit in which the radius of one of the two circles can be considered large we want to recover the symplectic form of the cotangent space, which is

ω = dα ∧ dβ. (1)
This indicates immediately what we need:

ω = -h -1 dh ∧ k -1 dk. ( 2
)
which locally is just the same as the previous one. The corresponding Poisson brackets are easily computed:

{k, h} = hk. (3)
This defines the phase space. Let us look for a corresponding quantum theory. For this, we want a Hilbert space H and operators h and k with an operator algebra that reduces to the above Poisson algebra in the appropriate limit. The problem is easy to solve. The Hilbert space is the finite dimensional Hilbert space H with a discrete basis |n where n = 1, ..., N = dim H, and the operators act as follow

k|n = e i 2π N n |n (4)
and

h|n = |n + 1 (5)
cyclically (that is k|N = |1 ). A straightforward calculation gives their commutator algebra

[h, k] = e i 2π N -1 hk. (6)
which gives a representation of the Poisson algebra for large N . In this limit, the Planck constant (determined by

[â, b] = i {a, b}) is related to N by = 2π N . (7)
To understand the physics, recall that we have a preferred length here: the constant curvature radius, which we have set to unit. Therefore the dimensionless quantity 2π N is actually the ratio between two dimensionfull quantities: in the physical theory it is the ratio of the Planck length scale to the cosmological constant scale.

The physics of the quantum theory of this simple example is intriguing. Since the phase space is compact, the Hilbert space is finite dimensional and therefore both h and k have discrete spectrum. This is like having a particle on a circle, and therefore discrete momentum, but also the circle being actually discrete, and therefore discrete position. Discreteness of the position gives a maximum momentum. So, all physical quantities are discrete, bounded and therefore completely finite. This is appealing. The mathematics of the continuum is just a useful approximation to a physical reality which is always discrete and finite.

V. SU(2) × SU(2)

Our task is now to repeat the previous exercise for Γ = SU (2) × SU (2). We use the notation

(k = e J , h) ∈ SU (2) × SU (2), (8)
In the R → ∞ limit where the length of the arc is small compared to R, we have k ∼ 1 1 + J. The symplectic structure of T * SU (2) is defined by the symplectic form ω = dθ where

θ = T r[Jh -1 dh]. (9)
There are several possibilities for deforming this structure to have it well defined on SU (2) × SU (2). The simplest possibility is to take

θ = T r[kh -1 dh] (10)
which reduces to (9) in the limit. The resulting symplectic form

ω = T r[dk ∧ h -1 dh -kh -1 dh ∧ h -1 dh] (11)
is closed and invariant under h → λh with λ ∈ SU (2). (It is not invariant under k → λk, but this transformation is not a symmetry: it transforms small triangles into large ones). But there are others and more interesting phase space structures that one can write on the group SU (2) × SU (2). The different forms of compatibility between phase space structure and group structure are studied under the name of Poisson-Lie groups and quasi Poisson-Lie groups [START_REF] Meusburger | Quaternionic and Poisson?Lie structures in three-dimensional gravity: The cosmological constant as deformation parameter[END_REF][START_REF] Marmo | Poisson Lie Group Symmetries for the Isotropic Rotator[END_REF][START_REF] Lu | Multiplicative and Affine Poisson Structures on Lie Groups[END_REF][START_REF] Alekseev | Manin Pairs and Moment Maps[END_REF][START_REF] Alekseev | Quasi-Poisson Manifolds[END_REF][START_REF] Schroers | Quantum gravity and non-commutative spacetimes in three dimensions: a unified approach[END_REF].

Instead of trying to guess physically interesting symplectic (or Poisson) structures, we go directly to the quantum theory and study the problem there. We can then recover a phase space structure form the classical limit of the quantum operator algebra. After all, in the real world it is the quantum theory to have a classical limit, not the other way around.

What we are seeking is therefore a deformation of the standard LQG operator algebra giving a finite Hilbert space. For this, let us start recalling the action of the operators h and k = e J on the standard LQG representation. This is,

hψ(U ) = U ψ(U ) ( 12
)
and

J i ψ(U ) = L i ψ(U ). ( 13
)
where L i is the left invariant derivative operator on the group manifold. Let us transform this to the canonical basis of

L 2 [SU (2)] ∼ ⊕ ∞ j=0 (H j ⊗ H j ) U |jmn = D j mn (U ) (14)
defined by the Wigner matrices D j mn . The result is

J i |jmn = τ (j) mk |jkn . ( 15
)
where τ

(j)
mk are the SU (2) generators in the j representation, and

h AB |jmn = 1 2 j j A m m 1 2 j j B n n |j m n (16)
where A, B are the indices of the SU (2) matrix, repeated indices are summed over and the matrices are the Wigner 3j symbols. Equation ( 16) is obtained from ( 12) and ( 14) by noticing that the group elements are the same as their spin-1 2 representation and using the standard decomposition of products of representations.

The first of these two equations is analogous to the abelian case: the operator is defined on the finite dimensional Hilbert space formed by a single spin component.

The second equation, however, requires the quantum state space to be infinite dimensional, because there are non-vanishing Wigner 3j symbols anytime j = j ± 1 2 . Can this equation be modified to adapt it to a finite dimensional Hilbert space? The answer is well known: let us define the action of the h operator in the constant curvature case to be

h AB |jmn = 1 2 j j A m m q 1 2 j j B n n q |j m n (17)
where we have replaced the Wigner 3j symbols with their q deformation, with q r = -1 for an integer r [START_REF] Kauffman | Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds[END_REF]. These operators are now well defined on the finite dimensional Hilbert space

H = ⊕ jmax j=0 (H j ⊗ H j ) (18)
where

j max = r -2 2 . ( 19
)
Equations ( 15) and ( 17) define the quantum theory in the constant curvature case, and represent our main proposal for the kinematics of quantum gravity in the presence of a cosmological constant. (Compare also with [START_REF] Major | On the q-quantum gravity loop algebra[END_REF].)

The h operators no longer commute. This can be seen using the graphical notation. Writing the Wigner symbols as a trivalent node, the matrix elements of the h operator read

(h AB ) mn m n = . ( 20
)
If we act with two operators we have

h AB h CD = . ( 21
)
while acting in the reverse order gives

h CD h AB = . ( 22
)
In the R → ∞ (or q = 1) case, the crossing of two lines gives at most a sign, which squared gives unit. Therefore the two operators commute. In the q deformed case, the crossing gives a q dependent factor [START_REF] Kauffman | Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds[END_REF][START_REF] Roberts | Skein theory and Turaev-Viro invariants[END_REF] and therefore the operators fail to commute. If we call R the operator giving the crossing, we have

h AB h CD = R A C AC R B D BD h C D h A B , (23)
and if R can be expanded in as R ∼ 1 + r, we obtain classical Poisson brackets of the form

{h AB , h CD } = r B D BD h CD h AB + r A C AC h C D h A B . ( 24
)
Since the h operators do not commute, there is no h representation in the quantum theory anymore. This completes the quantization of the SU (2) × SU (2) phase space. As in the abelian case, the resulting Hilbert space is finite dimensional. The dimension of the Hilbert space is determined by the ratio between the two constants: the one introduced by the quantization (physically: the Planck constant scale), and the constant curvature of the simplices, which enters via the deformation of the Poisson algebra -physically: the cosmological constant.

A q-deformation of the dynamics renders quantum gravity finite. This is known since the early nineties in 2+1 dimensions thanks to the Turaev-Viro state sum model [START_REF] Turaev | State sum invariants of 3 manifolds and quantum 6j symbols[END_REF], which renders Ponzano Regge 2+1 quantum gravity finite, and whose strict connection to LQG was early pointed out [START_REF] Rovelli | The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis[END_REF]. The result has been extended to 3+1 dimensions in [START_REF] Fairbairn | Quantum deformation of two four-dimensional spin foam models[END_REF][START_REF] Han | 4-dimensional Spin-foam Model with Quantum Lorentz Group[END_REF][START_REF] Han | Cosmological Constant in LQG Vertex Amplitude[END_REF][START_REF] Han | Covariant Loop Quantum Gravity, Low Energy Perturbation Theory, and Einstein Gravity[END_REF]. The q-deformation of the dynamics amounts to the introduction of a cosmological constant [START_REF] Smolin | Linking topological quantum field theory and nonperturbative quantum gravity[END_REF][START_REF] Mizoguchi | Three-dimensional gravity from the Turaev-Viro invariant[END_REF]; in 2+1, the relation between the deformation parameter q and the cosmological constant λ is [START_REF] Mizoguchi | Three-dimensional gravity from the Turaev-Viro invariant[END_REF][START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF]

q = e i √ Λ G . (25)
A q deformations has been derived in LQG as a way to implement the dynamics of the theory with cosmological constant in [START_REF] Noui | Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity[END_REF][START_REF] Pranzetti | Turaev-Viro amplitudes from 2+1 loop quantum gravity[END_REF]. Here we have shown that a q-deformation is also directly implied by the constant curvature of the individual simplices, and we have given the corresponding form of the geometrical operators of the gravitational theory.

VI. PHYSICAL CONSIDERATIONS

The Hilbert space H constructed above reduces to the usual LQG Hilbert space when the triangles are small compared to the curvature radius. Namely when the region considered is small compared to the cosmological constant scale. But there is a bound to this smallness, which is determined by the value of j max , namely by the ratio of the cosmological constant scale to the Planck scale. The region considered can never be smaller than the Planck scale and therefore never arbitrarily small, in the units in which the constant curvature radius is units. (See [START_REF] Bianchi | A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity[END_REF] for a discussion of the geometrical interpretation of this ratio.)

The effect of the compactness of SU (2) in conventional LQG is the discretization of the intrinsic geometry. In the quantization considered here, which takes the cosmological constant into account, there is a further compactness: the entire phase space, and not just the configuration space, is compact. Therefore also the variable conjugate to the intrinsic geometry is compact. This is evident form the fact that Hilbert space is finite dimensional (for each link), and therefore all local operators have discrete spectrum. Therefore the extrinsic geometry is quantized as well.

The extrinsic curvature K ab determines the rate of change of the intrinsic geometry, because (in the Lapse=1, Shift=0 gauge) it is the proper-time derivative of the metric: K ab ∼ dq ab /dt. Since q ab (∆t) ∼ q ab (0) + dq ab /dt ∆t, we can infer the lapsed proper time ∆t from the values of q ab (0), q ab (∆t) and K ab . Since all these quantities have discrete spectrum, we expect proper-time intervals, measured using gravitational observables, to be discrete as well. While one expects the scale of minimum proper time to be Planckian, its full discrete spectrum can depend on the cosmological constant. This is similar to the angle discreteness pointed out by Seth Major [START_REF] Major | Quantum Geometry Phenomenology: Angle and Semiclassical States[END_REF]. Do these results generalise to 3+1 dimensions? In the Euclidean case, the situation appears very similar. In the presence of a cosmological constant, we cannot choose a discretization of spacetime with flat simplices, because these are not solutions of the field equations. A constant curvature four simplex is bounded by constant curvature tetrahedra, and the geometry of these determines again a compact space, as for the curved triangles considered above. A compact space, in turn, determines a finite Hilbert space for each link.

In the Lorentzian case, the problem is more subtle, because of the hyperbolic geometry. However, it seems reasonable to require single cells of the triangulation to stay within the De Sitter horizon. This again yields a maximal size, determined by the cosmological constant, and therefore a finite Hilbert space per each link. The situation, however, is still unclear in the Lorentzian 3+1 case. ∼∼∼ Note: We understand that Han, Haggard, Kamińsky and Riello have related results in 3+1 dimensions, which will appear soon.

Acknowledgements.

FV thanks Hal Haggard and Muxin Han for detailed explanations on [8], before publication, which have inspired this works. CR thanks Maité Dupius, Florian and Ela Girelli, and Aldo Riello, for extensive discussion on the topic. FV acknowledges support from the Netherlands Organisation for Scientific Research (NWO) Veni Fellowship Program. CR acknowledges support from Samy Maroun Research Center for Time, Space and the Quantum.


F Candelier

J Einarsson

B Mehlig

Angular dynamics of a small particle in turbulence

Keywords: numbers: 05.40.-a, 47.55.Kf, 47.27.-i, 92.60.Mt

We compute the angular dynamics of a neutrally buoyant nearly spherical particle immersed in an unsteady fluid. We assume that the particle is small, that its translational slip velocity is negligible, and that unsteady and convective inertia are small perturbations. We derive an approximation for the torque on the particle that determines the first inertial corrections to Jeffery's equation. These corrections arise as a consequence of local vortex stretching, and can be substantial in turbulence where local vortex stretching is strong and closely linked to the irreversibility of turbulence.

The angular dynamics of small non-spherical particles in flows is often discussed in terms of Jeffery's theory [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF][START_REF] Parsa | Rotation rate of rods in turbulent fluid flow[END_REF][START_REF] Gustavsson | Tumbling of small axisymmetric particles in random and turbulent flows[END_REF][4], neglecting the effects of particle and fluid inertia. It is assumed that the instantaneous torque on the particle vanishes at every instant in time, so that the angular velocity of a small spherical particle equals half of the fluid vorticity at the particle position. But this is no longer true when the particle is so large that inertial effects become important. It is straightforward to take into account particle inertia [START_REF] Gustavsson | Tumbling of small axisymmetric particles in random and turbulent flows[END_REF][START_REF] Marchioli | Orientation, distribution, and deposition of elongated, inertial fibers in turbulent channel flow[END_REF][START_REF] Challabotla | Orientation and rotation of inertial disk particles in wall turbulence[END_REF][START_REF] Voth | Disks aligned in a turbulent channel[END_REF]. This may be a good approximation for heavy particles, but for neutrally buoyant particles the acceleration of the surrounding fluid cannot be neglected. How to model the effect of fluid inertia on the angular motion of a suspended particle is an important question [4,[START_REF] Voth | Disks aligned in a turbulent channel[END_REF], yet difficult to answer. Past theoretical studies have therefore often concentrated on simple cases, for example on the angular dynamics of axisymmetric particles in shear flows, analysing the stability of Jeffery orbits under inertial perturbations [START_REF] Saffman | On the motion of small spheroidal particles in a viscous liquid[END_REF][START_REF] Subramanian | Inertial effects on fibre motion in simple shear flow[END_REF][START_REF] Einarsson | Rotation of a spheroid in a simple shear at small Reynolds number[END_REF][START_REF] Rosén | Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers[END_REF].

In this Letter we estimate the first inertial contributions to the angular dynamics of a nearly spherical particle in a time-dependent and spatially varying flow. Our three most important assumptions are that the particle is small, almost neutrally buoyant, and nearly spherical (details are given below). We find that a spherical particle rotates at an angular velocity different from the local flow vorticity. The difference is caused by local vortex stretching. For nearly spherical particles we find additional contributions that depend in a more complex manner on the local fluid-velocity gradients. In turbulence these effects can alter the angular particle dynamics substantially. Recently the question was raised how the dynamics of tracer particles reflects the irreversibility of turbulence [START_REF] Xu | The pirouette effect in turbulent flows[END_REF][START_REF] Jucha | Timereversal-symmetry breaking in turbulence[END_REF][START_REF] Xu | Flightcrash events in turbulence[END_REF][START_REF] Pumir | Single-particle motion and vortex stretching in threedimensional turbulent flows[END_REF]. Our results show how the inertial angular dynamics of a small particles is linked to the breaking of time-reversal invariance in turbulence.

Formulation of the problem. Consider a small, nearly neutrally buoyant, spheroidal particle (with rotational symmetry axis n) in a fluid. We assume that the particle is nearly spherical, its aspect ratio λ is close to unity: λ ≡ 1 + where is a small parameter determining the eccentricity of the particle: > 0 for prolate particles, while < 0 for oblate particles. We assume that the particle is small: κ ≡ 2a/ 1. Here 2a is the length of the symmetry axis of the particle, and is the length over which the flow can be linearised near the particle.

We assume that inertial effects matter, but that they are weak. Convective inertia due to the fluid-velocity gradients is characterised by the shear Reynolds number Re s ≡ a 2 s/ν, where ν is the kinematic viscosity of the fluid, and s measures the magnitudes of the fluid-velocity gradients, the strain rate. The effect of unsteady inertia depends on the time scale τ c that describes how fast the boundary conditions change. The ratio of the magnitude of unsteady and convective inertia defines the Strouhal number Sl ≡ (sτ c ) -1 . The effect of particle inertia on the angular dynamics is determined by the Stokes number, the ratio of the rate of change of angular momentum and the torque: St ≡ (ρ p a 5 s)/(τ c µsa 3 ) = (ρ p /ρ f )Re s Sl. Here ρ f and ρ p are the mass densities of the fluid and the particle, and µ = ρ f ν is the dynamic viscosity.

We treat the effect of inertia perturbatively, this requires Re s and Re s Sl to be small (but not too small, see below). We disregard the effect of translational slip. This is justified for small particle Reynolds number, Re p ≡ av s /ν Re 1/2 s . Here v s is the slip velocity. To ensure that it is small enough we assume that the particle is approximately neutrally buoyant.

Equations of motion. The equations that govern the angular particle dynamics read:

dn dt = ω ∧ n , I • dω dt + dI dt • ω = T . ( 1
)
Here I is the moment-of-inertia tensor of the particle, with elements I ij = A I n i n j + B I (δ ijn i n j ), A I and B I are the moments of inertia around and transverse to the axis n, and T is the hydrodynamic torque:

T = Sp r ∧ σ • ds . ( 2
)
The integral is over the particle surface S p , ds is the outward normal surface element, and σ(x, t) is the stress tensor of the fluid at position x, and r ≡ x-x p where x p is the particle position. The torque ( 2) is determined by the solution of Navier-Stokes equations. We decompose the stress as σ = σ ∞ + σ (1) , where σ ∞ is the stress tensor of the undisturbed Eulerian fluid velocity, denoted by U ∞ (x, t). The second term, σ (1) , is the contribution to the stress tensor from the disturbance flow. The torque is decomposed in a similar way, T = T ∞ + T (1) . We compute these two contributions to the torque separately.

Torque due to stress of undisturbed flow. The undisturbed Eulerian fluid velocity U ∞ (x, t) satisfies Navier-Stokes equations in the laboratory frame:

∇ • σ ∞ = ρ f (∂ t U ∞ ) x + (U ∞ •∇)U ∞ ≡ ρ f D Dt U ∞ . (3)
The last equality defines the Lagrangian derivative along U ∞ , ∇ denotes the spatial derivative with respect to x, and the partial time derivative is evaluated at fixed x. The torque due to the undisturbed stress can be expressed as a volume integral using Eq. (3):

T ∞ = ρ f Vp dv r ∧ DU ∞ Dt . (4)
To evaluate this expression further we expand DU ∞ (x, t)/Dt around the particle position x p :

DU ∞ k Dt x(t) = DU ∞ k Dt xp(t) + r m ∂ ∂x m DU ∞ k Dt xp + r m r n ∂ ∂x m ∂ ∂x n DU ∞ k Dt xp(t) + O r 3 / 3 . (5)
The components are relative to a fixed Cartesian basis in the laboratory frame. Substituting this expansion into Eq. ( 4) we find:

T ∞ i = ρ f Vp d 3 r ε ijk r j ∂ ∂x m DU ∞ k Dt xp r m + O r 4 / 4 . ( 6
)
The order is O(r 4 / 4 ) because terms odd in r vanish upon integration. The partial derivative in the integrand of Eq. [START_REF] Challabotla | Orientation and rotation of inertial disk particles in wall turbulence[END_REF]

evaluates to ∇(DU ∞ /Dt)| xp = (DA ∞ /Dt)| xp + A ∞ p • A ∞ p ,
where the elements of A ∞ p are the gradients of the undisturbed fluid velocity U ∞ at the particle centre:

(A ∞ p ) ij ≡ ∂U ∞ i /∂x j | x=xp .
Finally, to perform the integral in Eq. ( 6) we use the definition of the moment-of-inertia tensor. We obtain:

T ∞ = ρ f ρ p I • DΩ ∞ Dt xp -S ∞ p • Ω ∞ p (7) +(A I -B I ) S ∞ p • S ∞ p +O ∞ p • O ∞ p + DS ∞ Dt xp • n ∧ n .
Here we have decomposed the matrix A ∞ p into its symmetric and antisymmetric parts, the strain-rate matrix S ∞ p , and

O ∞ p . The matrix O ∞ p is linked to Ω ∞ p ≡ 1 2 ∇ ∧ U ∞ through O ∞ • r = Ω ∞ ∧ r.
Eq. ( 7) is valid for a spheroid with arbitrary aspect ratio (provided that κ 1). FIG. 1. Schematic of vectors used to describe the particle motion, x is a position in the laboratory frame, O is its origin, and xp(t) is the particle position at time t. The perturbation theory leading to Eq. ( 13) assumes that the particle follows the Lagrangian fluid trajectory.

Using the vorticity equation

DΩ ∞ Dt -S ∞ •Ω ∞ = ν∇ 2
Ω ∞ evaluated at the particle position, we can express the difference in the first row of Eq. ( 7) in terms of the Laplacian of Ω ∞ which vanishes for a strictly linear flow. In general, however, the non-linearity of U ∞ (x, t) results in a non-zero value of ∇ 2 Ω ∞ at the particle position.

Disturbance torque. We calculate the torque due to the disturbed fluid in perturbation theory, assuming that inertial effects are small and neglecting translational slip. This dictates how the problem must be de-dimensionalised: t = τ c t , r = ar , U = saU , and σ = µsσ . In the remainder of this Letter we use these dimensionless variables. To simplify the notation we drop the primes, all equations below are written in dimensionless variables. The disturbance caused by the particle has the flow velocity w(r, t) ≡ U (r+x p , t)-U ∞ (r+x p , t). It is defined to be a function of r(t) ≡ xx p (t) (Fig. 1). In dimensionless variables the disturbance problem reads: ∇ r • σ (1) (8)

= Re s Sl(∂ t w) r +(A ∞ p •r)•∇ r w+A ∞ p •w+(w•∇ r )w , w(r, t) = -(A ∞ p • r -ω ∧ r) for r ∈ S p , w(r, t) = 0 as |r| → ∞ .
The partial time derivative is evaluated at fixed r, and we have linearised

U ∞ (x, t) around x p , U ∞ (x, t) = U ∞ (x p , t) + A ∞ p • r.
When is it justified to use this linear form in the disturbance problem (8)? The disturbance caused by the particle decays exponentially at distances larger than the Saffman length S ≡ 1/ √ Re s , so that we must require S < , where is the length scale over which the flow can be linearised. This condition is more restrictive than κ 1. In other words the shear Reynolds number Re s should not be too small, because convective inertia causes the disturbance to decay at S .

We use the reciprocal theorem [START_REF] Lovalenti | The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number[END_REF] to find the hydrodynamic torque on the particle, given an 'auxiliary' Stokes solution in the same geometry [START_REF] Subramanian | Inertial effects on fibre motion in simple shear flow[END_REF][START_REF] Einarsson | Rotation of a spheroid in a simple shear at small Reynolds number[END_REF][START_REF] Einarsson | Effect of weak fluid inertia upon Jeffery orbits[END_REF][START_REF] Candelier | The role of inertia for the rotation of a nearly spherical particle in a general linear flow[END_REF]. In dimensionless variables the reciprocal theorem reads:

T = T ∞ + T (0) -Re s V dv M T • f (w) . (9)
The first term on the r.h.s. is the torque due to the undisturbed fluid stresses, Eq. ( 7), expressed in dimensionless variables. The second term is Jeffery's torque [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF]:

T (0) = -K•(ω -Ω ∞ p )+H : S ∞ p ,
where K and H are Brenner's resistance tensors [START_REF] Kim | Microhydrodynamics: principles and selected applications[END_REF]. The third term in Eq. ( 9) is the torque due to the disturbance flow beyond the Stokes approximation. The tensor M(r) is determined by the known auxiliary Stokes solution (ũ = M • ω), the Stokes flow around the particle rotating with angular velocity ω in a quiescent fluid, see supplemental material [20]. The integration domain in [START_REF] Subramanian | Inertial effects on fibre motion in simple shear flow[END_REF] is the fluid volume outside the particle, and Re s f is defined as the r.h.s of the first Equation [START_REF] Saffman | On the motion of small spheroidal particles in a viscous liquid[END_REF].

To simplify the calculation of the third term in Eq. ( 9) we assume that the particle is nearly spherical [START_REF] Candelier | The role of inertia for the rotation of a nearly spherical particle in a general linear flow[END_REF], λ = 1 + , and expand in the small parameter . To order 2 , for example, the moments of inertia around and transverse to the particle symmetry axis [START_REF] Candelier | The role of inertia for the rotation of a nearly spherical particle in a general linear flow[END_REF] read in dimensionless variables:

A I = 8π 15 (1 -4 + 6 2 ) and B I = 8π 15 (1-3 + 7 2
2 ). For small values of the contribution of the volume integral in Eq. ( 9) can be evaluated. Details are given in the supplemental material [20].

For a spherical particle ( = 0) we find to order O(Re s ):

T = -8π(ω-Ω ∞ p )+8πRe s Sl 15 DΩ ∞ Dt xp -2 5 S ∞ p •Ω ∞ p . (10)
We now use Eqs. ( 10) and (1) to determine the particle angular velocity ω to order O(Re s ). For a spherical particle I has the elements I ij = 8πδ ij /15, in dimensionless variables. In perturbation theory dω/dt = dΩ ∞ p /dt = DΩ ∞ /Dt| xp . We must also require that ρ p ≈ ρ f , so that Re p remains small. To order O(Re s ) we find:

ω = Ω ∞ p + 1 15 (Re s Sl -St) DΩ ∞ Dt xp -2 5 Re s S ∞ p •Ω ∞ p . (11)
Eq. ( 11) is the main result of this Letter. We see that a very small particle in a viscous flow rotates with half the fluid vorticity, Ω ∞ p , as expected. The first inertial correction term, proportional to DΩ ∞ /Dt| xp , resembles the form of the slip velocity of a small particle subject to particle inertia and to the force due to the undisturbed pressure gradients. A small-St expansion gives

v s = St (ρ f /ρ p -1) DU ∞ /Dt| xp (12)
for Sl = 1. Maxey used this approximation to conclude that heavy particles are centrifuged out of vortical regions in turbulence [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF][START_REF] Gustavson | Statistical models for spatial patterns of heavy particles in turbulence[END_REF]. The expression for v s is of the same form as the second term on the r.h.s. of Eq. ( 11), since Re s = (ρ f /ρ p )St for Sl = 1. This term predicts that a particle that is slightly heavier than the fluid rotates a little bit more slowly, because it cannot keep up with the the fluid acceleration. A lighter particle, by contrast, rotates faster than Ω ∞ p . But these arguments disregard the S ∞ p •Ω ∞ p -term in Eq. ( 11). This inertial term connects the angular particle dynamics to vortex stretching in the undisturbed flow. This term vanishes in steady linear 13), along a Lagrangian path in turbulence (DNS at Re λ = 433 using JHU turbulence database [START_REF] Li | A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence[END_REF][START_REF] Yu | A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence[END_REF]). b Vorticity along the same path. c Distribution of Y (DNS, lines). Also shown: experimental data (•) at Re λ = 50 read off from Fig. 2d in Ref. [START_REF] Guala | On the evolution of material lines and vorticity in homogeneous turbulence[END_REF], rescaled by τK ≡ Tr A T • A -1/2 , see text. The experimental value (0.27 s -1 ) was taken from Ref. [START_REF] Lüthi | Lagrangian measurement of vorticity dynamics in turbulent flow[END_REF].

Y (t) |Ω ∞ p (t)| 2 τ 2 K t/τK Y P (Y ) a b c FIG. 2. a Inertial correction Y ≡ τKΩ ∞ p • S ∞ p • Ω ∞ p /|Ω ∞ p | 2 , Eq. (
flows such as a simple shear [START_REF] Lin | Simple shear flow around a rigid sphere: inertial effects and suspension rheology[END_REF][START_REF] Meibohm | Angular velocity of a sphere in a simple shear at small Reynolds number[END_REF], and for planar flows because vorticity is orthogonal to the flow plane.

Angular velocity and vortex stretching in turbulence. Fully developed turbulent flows exhibit large local vortex stretching rates, this is how the turbulent kinetic energy is dissipated at small scales. Irreversibility of turbulence [START_REF] Xu | The pirouette effect in turbulent flows[END_REF][START_REF] Jucha | Timereversal-symmetry breaking in turbulence[END_REF][START_REF] Xu | Flightcrash events in turbulence[END_REF][START_REF] Pumir | Single-particle motion and vortex stretching in threedimensional turbulent flows[END_REF] implies that the stretching rate

Ω ∞ • (S ∞ • Ω ∞ )
does not average to zero [START_REF] Guala | On the evolution of material lines and vorticity in homogeneous turbulence[END_REF][START_REF] Johnson | Large-deviation statistics of vorticity stretching in isotropic turbulence[END_REF]. This matrix element is positive on average because Ω ∞ tends to align with the middle eigenvector of S ∞ [START_REF] Xu | The pirouette effect in turbulent flows[END_REF], and its eigenvalue is positive on average. Eq. [START_REF] Rosén | Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers[END_REF] shows that the inertial correction to the angular velocity of a neutrally buoyant sphere in turbulence is determined by the stretching rate,

|ω| 2 /|Ω ∞ p | 2 ≈ 1 -4 5 Re s Ω ∞ p • (S ∞ p • Ω ∞ p )/|Ω ∞ p | 2 , ( 13
) and thus linked to the breaking of time-reversal invariance in turbulence.

But are the conditions of validity summarised above met for a neutrally buoyant particle in turbulence? The shear rate, defined by s = Tr S • S 1/2 , is of the same order as τ -1 K , the inverse of the Kolmogorov time τ K ≡ Tr A T • A -1/2 . This means that the shear Reynolds number is of order Re s ∼ a 2 /η 2 K , where η K = √ ντ K is the Kolmogorov length. We conclude that the particle must be smaller than the Kolmogorov length for the perturbation theory in Re s to be valid. For a nearly neutrally buoyant particle we can disregard gravity, and we assume that no other external forces act on the particle. Therefore the timescale τ c is that of the fluid and we take τ c = s -1 ∼ τ K , so that the Strouhal number is unity. The perturbation theory also requires that the Saffman length S ≡ a/ √ Re s ∼ η K is smaller than the length over which the flow can be linearised. In turbulence ∼ 10η K . So this condition is only marginally satisfied. Finally, translational slip is negligible when the Oseen length is much larger than the Saffman length. This is the case when Re 2 p Re s . For small neutrally buoyant particles in turbulence this condition is well satisfied.

We evaluate the inertial correction in Eq. ( 13) for a neutrally buoyant particle numerically, following a Lagrangian trajectory in fully developed turbulence. We use the JHU turbulence database [START_REF] Li | A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence[END_REF][START_REF] Yu | A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence[END_REF] that contains a direct numerical simulation of forced, isotropic turbulence at Re λ = 433. The result is shown in Fig. 2a. Panel b shows the vorticity along the same path. We observe the stretching of a vortex tube at t ≈ 20 τ K . We see that the inertial correction to the particle angular velocity can be substantial during vortex stretching, a factor of order unity times Re s . Panel c shows that the distribution of the inertial correction has heavy, non-Gaussian tails that give rise to large values of the inertial correction. Comparison with experimental data at Re λ = 50 (•), from Fig. 2d in Ref. [START_REF] Guala | On the evolution of material lines and vorticity in homogeneous turbulence[END_REF], shows that the tails are quite robust even at moderate values of Re s , a consequence of the universality of dissipative-range turbulent fluctuations [START_REF] Schumacher | Small-scale universality in fluid turbulence[END_REF].

Non-spherical particles. We have computed the angular velocity also for nearly spherical particles (| | 1). Neglecting inertial effects we obtain an -expansion of Jeffery's equation [START_REF] Jeffery | The motion of ellipsoidal particles immersed in a viscous fluid[END_REF]:

ω = Ω ∞ p + ( + 2 2 )(n ∧ S ∞ p • n) + O( 3
). The form of the first inertial corrections to Jeffery's angular velocity is constrained by symmetries. To linear order in St and Re s the corrections are quadratic in S ∞ p and Ω ∞ p , and linear in time derivatives of S ∞ p and Ω ∞ p . The inertial corrections to ω must be invariant under n → -n, and S ∞ p is symmetric and traceless. Only the following terms can occur in the O( Re s )-correction δω ≡ Re s ω ( Res) + Stω ( St) to Eq. ( 11):

δω = β 1 DΩ ∞ Dt xp +β 2 ( DΩ ∞ Dt xp • n)n + β 3 dS ∞ p dt • n ∧ n (14) +β 4 S ∞ p • Ω ∞ p +β 5 [n • (S ∞ p • Ω ∞ p )]n+β 6 [(n • (S ∞ p • n)]Ω ∞ p +β 7 (n • Ω ∞ p )(Ω ∞ p ∧ n) + β 8 (S ∞ p • S ∞ p • n) ∧ n +β 9 [S ∞ p • (Ω ∞ p ∧ n)] ∧ n + β 10 (n • Ω ∞ p ) S ∞ p • n .
For Sl = 1 we find using the method described above (details in the supplemental material [20]):

β 1 = -3 15 (Re s -St) , β 2 = -15 (Re s -St) ( 15
)
β 3 = -3 (Re s -1 5 St) , β 4 = 733 525 Re s , β 5 = -15 ( 4 35 Re s + St) , β 6 = -15 ( 30 35 Re s -St) , β 7 = -15 (Re s -St) , β 8 = -8 21 Re s , β 9 = -15 (4Re s -St) , β 10 = 3
35 Re s . Eq. [START_REF] Xu | Flightcrash events in turbulence[END_REF] shows that the inertial corrections to the angular velocity of non-spherical particles depend intricately on the relative alignment of the particle symmetry axis, of the vorticity, and of the eigensystem of the strain-rate matrix [START_REF] Parsa | Rotation rate of rods in turbulent fluid flow[END_REF][START_REF] Gustavsson | Tumbling of small axisymmetric particles in random and turbulent flows[END_REF][START_REF] Byron | Shape-dependence of particle rotation in isotropic turbulence[END_REF]. For a neutrally buoyant particle in incompressible isotropic homogeneous turbulence we can average the correction. Since δω contains a factor , we obtain the O( Re s )-result by averaging n uniformly over the unit sphere, n i n j = δ ij /3. This gives:

Ω ∞ p • δω = (4/3) Re s Ω ∞ p • (S ∞ • Ω ∞ p )
. On average the form of the correction is similar to that in Eq. [START_REF] Rosén | Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers[END_REF]. We see that the inertial effect is weakened for slightly prolate neutrally buoyant particles, but increased for oblate particles.

Conclusions. We have computed the first inertial corrections to the angular velocity of a small, approximately neutrally buoyant particle in a space-and timedependent flow. Our main prediction, Eq. ( 13), expresses the inertial correction to the angular velocity of a small sphere in terms of a matrix element that determines vortex stretching. This shows that the inertial angular dynamics of a small neutrally buoyant sphere in turbulence picks up that time-reversal invariance is broken [START_REF] Xu | The pirouette effect in turbulent flows[END_REF][START_REF] Jucha | Timereversal-symmetry breaking in turbulence[END_REF][START_REF] Xu | Flightcrash events in turbulence[END_REF][START_REF] Pumir | Single-particle motion and vortex stretching in threedimensional turbulent flows[END_REF].

Our results demonstrate that convective and unsteady fluid inertia must be treated on an equal footing in turbulence. It is commonly argued that convective effects can be disregarded when the translational slip velocity v s is negligible. But we have shown here that substantial inertial corrections may arise from convective terms due to turbulent strains. Such terms are likely to be important in the translational problem too, so that the 'Maxey-Riley' equations [START_REF] Gatignol | The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow[END_REF][START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] cannot be used to describe the translational dynamics of small spheres in turbulence.

Particle-tracking experiments [START_REF] Traugott | Experimental study of forces on freely moving spherical particles during resus-pension into turbulent flow[END_REF][START_REF] Variano | [END_REF][START_REF] Marcus | Measurements of the solid-body rotation of anisotropic particles in 3d turbulence[END_REF] and particleresolving DNS required to test the predictions in this Letter have recently become possible [START_REF] Homann | Finite-size effects in the dynamics of neutrally buoyant particles in turbulent flow[END_REF][START_REF] Fornari | Reduced particle settling speed in turbulence[END_REF], but they are still very challenging. In order to stimulate the substantial effort required for these measurements we now give a concrete suggestion for how Eq. ( 13) could be tested in either experiment or particle-resolving DNS. First, the particle should be neutrally buoyant, and its size a must be smaller than the Kolmogorov length η K . Small a ensures that Re 2 p Re s since Re p ∝ a 3 [Eq. ( 12)] while Re s ∝ a 2 . This may be a difficult technical requirement. It is easier to meet when the turbulence intensity is low, so that η K is larger, and Fig. 2c shows that the effect persists for lower turbulence intensities. Second, to determine the vorticity of the undisturbed flow, one must measure and interpolate the flow near the particle [START_REF] Traugott | Experimental study of forces on freely moving spherical particles during resus-pension into turbulent flow[END_REF]. Then we suggest to consider the distribution of |ω| 2 /|Ω ∞ p | 2 -1. The width of this distribution must approach zero for a perfect tracer particle. Fig. 2c shows that the first effect of inertia is to substantially widen the tails of this distribution, and to slightly shift its mean value. Finally, our results for nearly spherical particles indicate that disks may be more sensitive to inertial corrections than rods.

Can our results be generalised to cases where v s is not negligible? An important case is settling [START_REF] Gustavsson | Clustering of particles falling in a turbulent flow[END_REF][START_REF] Ireland | The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects[END_REF][START_REF] Mathai | Microbubbles and microparticles are not faithful tracers of turbulent acceleration[END_REF][START_REF] Parishani | Effects of gravity on the acceleration and pair statistics of inertial particles in homogeneous isotropic turbulence[END_REF]. The settling of ice crystals, for instance, is important for rain initiation in cold turbulent clouds [START_REF] Pruppacher | Microphysics of clouds and precipitation[END_REF]. For a spatially constant flow the effect of non-zero v s was analysed by Lovalenti & Brady [START_REF] Lovalenti | The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number[END_REF]. Can one use their methods to compute lift forces [START_REF] Zimmermann | Rotational intermittency and turbulence induced lift experienced by large particles in a turbulent flow[END_REF] on small particles in turbulence? This is a difficult problem because it requires singular perturbation theory [START_REF] Saffman | The lift on a small sphere in a slow shear flow[END_REF]. Finally, larger particles pose other problems: they sense inertial-range turbulent fluctuations [START_REF] Parsa | Inertial range scaling in rotations of long rods in turbulence[END_REF], and wakes may affect their dynamics [START_REF] Mathai | Wake-driven dynamics of finitesized buoyant spheres in turbulence[END_REF].

Acknowledgments. We thank Howard Stone for discus-sions and for making the preprint [START_REF] Stone | Inertial effects on the rheology of suspensions and on the motion of individual particles[END_REF] available to us that considers a related question (the steady-state limit of the problem considered here). This work was supported by Vetenskapsrådet [grant number 2013-3992], Formas [grant number 2014-585], and by the grant 'Bottlenecks for particle growth in turbulent aerosols' from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048. The numerical results in Fig. 2 use data from the JHU turbulence database [START_REF] Li | A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence[END_REF][START_REF] Yu | A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence[END_REF].

B. Sphere in an unsteady flow

As in the previous Section we assume that ω = Ω ∞ p + O(Re s ). But now S ∞ p and Ω ∞ p depend on time. In evaluating the contribution from the volume integral perturbatively we must again set ω = Ω ∞ p in w (0) . We find:

I = 0 . (S30)
For non-spherical particles I is not zero, see below. The remaining terms (J, K, and L) remain unchanged. The torque to the disturbance flow [Eq. [START_REF] Xu | The pirouette effect in turbulent flows[END_REF] in the Letter] now reads

T ∞ = Re s 8π 15 Sl DΩ ∞ Dt xp -S ∞ p • Ω ∞ p . ( S31
)
This gives

T = -8π(ω -Ω ∞ p ) + Re s 8π 1 15 Sl DΩ ∞ Dt xp - 2 5 S ∞ p • Ω ∞ p . ( S32
)
In the unsteady case the torque does not vanish, in general. To determine the angular velocity of the particle we must therefore solve Newton's equation, Eq. ( 1) in the Letter. For the sphere dI/dt vanishes, so that we have St

15 dω dt = -(ω -Ω ∞ p ) + Re s Sl 15 DΩ ∞ Dt xp - 2 5 S ∞ p • Ω ∞ p . ( S33
ω = Ω ∞ p + 1 15 (Re s Sl -St) DΩ ∞ Dt xp - 2 5 Re s S ∞ p • Ω ∞ p . (S34)
This is Eq. ( 12) in the Letter.

C. Nearly spherical particle in unsteady flow

Now we consider a nearly spherical particle in an unsteady flow. This case is quite similar to the previous one, except that now the particle is assumed to rotate with an angular velocity that is different from Ω ∞ p . For a nearly spherical particle the angular slip velocity ω s = ω -Ω ∞ p is of order as we shall see. This follows from Jeffery's theory.

As the auxiliary problem we use the Stokes flow of a nearly spherical particle rotating with angular velocity ω in a quiescent fluid. The solutions of the auxiliary problem (and of the Stokes problem for w (0) ) are found using perturbation theory in the small parameter . Details of the method are given in the appendix of Ref. [S7].

We note that the problem to be solved is more difficult for a non-spherical particle than for a sphere. For a sphere the locus of the particle surface remains invariant under rotation. This simplifies the problem substantially, because we can work with a basis that remains fixed in the laboratory frame. For a non-spherical particle we need to consider how the locus of the particle surface changes. This may be accomplished in a coordinate system that rotates with the particle. The time derivative of w (0) in the volume integral is evaluated at fixed r. This derivative must be transformed to the rotating coordinate system:

Sl ∂w ∂t r = Sl ∂w i ∂t r ê i (t) + ω ∧ w -[(ω ∧ r) • ∇] w . (S35)
The partial time derivative on the r.h.s. is evaluated at a fixed point r relative to an observer rotating with the particle, ê i (t) are basis vectors rotating with the particle, and w i are the components of w in that basis. We remark that the primes in Eq. (S35) do not denote dimensionless variables.

The remaining three terms are also integrated in the rotating coordinate system (in this system the locus of S p does not change with time). Evaluating the volume integral in this way we find:

V dv f • u = -8π -Sl 3 dω s dt -1 3 S ∞ p • Ω ∞ p + 1 8 S ∞ p • ω s • ω -8π 628 525 S ∞ p • Ω ∞ p -32 105 (n • S ∞ p • n)Ω ∞ p + 136 525 (n • S ∞ p • Ω ∞ p )n • ω -8π -3 5 S ∞ p • O ∞ p -3Sl 5 dS ∞ p dt + 3 35 O ∞ p • S ∞ p -123 280 S ∞ p • S ∞ p • n ∧ n • ω . ( S36
)
Tr S ∞ = 0 Incompressibility S ∞ = (S ∞ ) T Symmetry O ∞ = -(O ∞ ) T Antisymmetry ω(n) = ω(-n) Inversion symmetry
In deriving this expression we neglected terms of the order of ω s and ω 2 s . This is justified because we know that the slip angular velocity is of order in the creeping-flow limit, see below. Setting = 0 in Eq. (S36) we find again Eq. (S17) derived in the previous Section using a different method (coordinate system fixed in the laboratory frame). We note that ω s = 0 to lowest order in Re s for = 0.

Returning to the non-spherical case we compute the angular velocity using the ansatz: ω = ω (0) + ω ( ) + 2 ω ( 2 ) + St ω (St) + Re s ω (Res) + St ω ( St) + Re s ω ( Res) . (S37)

To order O( ) 2 we obtain:

ω (0) + ω ( ) + 2 ω ( 2 ) = Ω ∞ p + + 1 2 2 n ∧ S ∞ p • n . (S38)
This is the second order of an expansion of Jeffery's equation [S1, S7] in the small parameter . Now consider the corrections due to particle and fluid inertia. For = 0 we find:

Re s ω (Res) + Stω (St) = 1 15 (Re s Sl -St) DΩ ∞ Dt xp -2 5 Re s S ∞ p • Ω ∞ p . ( S39
)
This is again Eq. (S34). For non-spherical particles there are corrections to this expression. To simplify the form of the corrections we use the symmetries of the problem summarised in Table I. We find that the O( Re s )-corrections to Eqs. (S38) and (S39) take the form 35 Re s . We see that the inertial correction to the angular velocity of a non-spherical particle depends intricately on the alignment of the vorticity Ω ∞ p (t) and the local eigensystem ŝj (t) of S ∞ p (t). In perturbation theory, to lowest order in Re s , all time derivatives are evaluated as Lagrangian time derivatives along fluid trajectories.

δω ≡ Re s ω ( Res) + St ω ( St) = β 1 DΩ ∞ Dt xp + β 2 n • DΩ ∞ Dt xp n + β 3 dS ∞ p dt • n ∧ n (S40) + β 4 S ∞ p • Ω ∞ p + β 5 [n • (S ∞ p • Ω ∞ p )]n + β 6 [(n • (S ∞ p • n)]Ω ∞ p + β 7 (n • Ω ∞ p )(Ω ∞ p ∧ n) + β 8 (S ∞ p • S ∞ p • n) ∧ n + β 9 [S ∞ p • (Ω ∞ p ∧ n)] ∧ n + β 10 (n • Ω ∞ p )
For a neutrally buoyant particle in incompressible isotropic homogeneous turbulence we can average the correction. Since δω contains a factor we obtain the correct result to order O( Re s ) by averaging n uniformly over the unit sphere, n i n j = δ ij /3.

Ω ∞ p • δω = 4 3 Re s Ω ∞ p • (S • Ω ∞ p ) . ( S42
)
This result is discussed in the Letter.

  S ∞ p • n . For Sl = 1 the coefficients read:β 1 = -3 15 (Re s -St) , β 2 = -15 (Re s -St) , β 3 = -3 (Re s -1 5 St) , β 4 = 733 525 Re s ,(S41) β 5 = -15 ( 4 35 Re s + St) , β 6 = -15 ( 30 35 Re s -St) , β 7 = -15 (Re s -St) , β 8 = -8 21 Re s , β 9 = -15 (4Re s -St) , β 10 = 3

  Using that Tr S • O = Tr O • O • O = 0 and Tr S•S = Tr O•O we find

)

  On the l.h.s. we can set dω/dt = dΩ ∞ p /dt = DΩ ∞ /Dt| xp . To order O(Re s ) this gives

TABLE I .

 I Symmetries used in deriving Eq. (S40).

Supplemental material for: Angular dynamics of a small particle in turbulence F. Candelier, 1 J. Einarsson All equations in this supplemental material are in dimensionless variables, t = τ c t , r = ar , U = saU , and σ = µsσ . For details see the Letter. For notational convenience we drop the primes in the following.

I. EVALUATION OF JEFFERY'S TORQUE

Eq. [START_REF] Einarsson | Rotation of a spheroid in a simple shear at small Reynolds number[END_REF] in the Letter gives an expression for Jeffery's torque [S1]:

written in terms of Brenner's resistance tensors K and H

Here

and S ∞ p is the symmetric part of the matrix A ∞ p of the undisturbed fluid-velocity gradients evaluated at x p . The anti-symmetric part is denoted by O ∞ p . This matrix is related to

Expanding the resistance tensors in the nearly-spherical limit one finds [S3]:

Here is a small parameter related to the particle eccentricity (see Ref. [S7]). For a sphere we have K ij = 8πδ ij and H = 0.

II. COMPUTATION OF DISTURBANCE TORQUE USING RECIPROCAL THEOREM

The hydrodynamic torque is given by

The integral is over the particle surface S p , and the vector ds points in the direction of the outward surface normal.

The stress tensor σ is determined by the solution of the Navier-Stokes equations. In dimensionless variables it has elements σ ij = -δ ij p + 2S ij where p is the pressure and S ij are the elements of S ∞ . We decompose σ as σ = σ ∞ + σ (1) , where σ ∞ is the undisturbed stress tensor and σ (1) is the contribution to the stress tensor from the disturbance flow. The hydrodynamic torque is split up in the same way:

We use the reciprocal theorem [S4] to compute T (1) , following Refs. [S5-S7]. The reciprocal theorem relates integrals of two different Newtonian and incompressible flows:

Here w is the disturbance flow, it obeys Eq. ( 9) in the Letter. The fields u and σ are the fluid velocity and the stress tensor of an 'auxiliary' Stokes problem in the same geometry, namely the Stokes flow produced by particle rotating with angular velocity ω in a quiescent fluid:

The hydrodynamic (auxiliary) torque on the particle is denoted by T :

The surface integrals in Eq. (S5) are over the particle surface, the vector ds points in the direction of the outward surface normal. The volume integrals are over the volume outside the particle. The surface integrals in Eq. ( S5) are evaluated by substituting the boundary conditions for the disturbance flow and the auxiliary flow on the particle surface. The volume integral on the l.h.s. of Eq. (S5) vanishes by virtue of Eq. (S6a).

For the divergence of the disturbance stress tensor in the volume integral on the r.h.s., we substitute Eq. ( 9) in the Letter. The vector f is defined by

We find:

The surface integrals can be expressed in terms of the torques T (1) and T :

This is an exact relation between the sought disturbance torque and the known auxiliary torque. The disturbance velocity w, however, is not known. To lowest order in Re s we can use the Stokes approximation w (0) in evaluating the volume integral. The function w (0) is the solution of the Stokes problem:

The first two terms on the r.h.s. of (S10) evaluate to ω • T (0) . Adding T ∞ [Eq. (S12)] we obtain Eq. ( 10) in the Letter, with u = M • ω.

III. EVALUATION OF TORQUE AND ANGULAR VELOCITY

A. Sphere in a steady flow

We first consider a simpler problem than that addressed in the Letter. What is the torque on a sphere in the steady state where the sphere rotates with the steady angular velocity ω = Ω ∞ p + O(Re s )? The undisturbed torque T ∞ follows from Eq. ( 8) in the Letter

The disturbance torque is evaluated using the reciprocal theorem (Section II). The solution of the auxiliary problem (S6) for a spherical particle reads:

The last equality defines the elements of the matrix M used in Eq. ( 8) in the Letter. The corresponding torque T evaluates to

and the surface integral in Eq. (S10) vanishes:

It remains to evaluate the volume integral in Eq. (S10). The Stokes solution (the solution of (S11b) for ω = Ω ∞ p ) reads:

The volume integral in Eq. (S10) consists of four terms that we evaluate separately:

Since the problem is steady, the derivative (∂w (0) /∂t) r vanishes, so that I is zero:

The integral J can be written in the form

The two integrals in this equation are evaluated using Maple:

This yields

To evaluate this expression further we split A ∞ p into its symmetric and anti-symmetric parts, A ∞ p = S ∞ p + O ∞ p , and make use of the relations

We use

The second equation follows from (O ∞ p ) ij = -ε ijk Ω k and ε ijk ε i m = δ j δ kmδ jm δ k . We finally obtain:

The remaining integrals (K and L) are more difficult to calculate. Using Maple we find:

Taking these results together we find:

This result is valid for an arbitrary choice of ω. It follows:

Adding the torque due to the undisturbed flow [Eq. (S12)] yields:

This is Eq. ( 11) in the Letter. In the steady state the torque must vanish. This gives