A Barrat

E Trizac

M H Ernst

Quasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases

Keywords: numbers: 45.70.-n, 05.20.Dd, 05.10.Ln, 02.70.Uu

The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation are studied in the QE-limit (Quasi-Elastic; infinitesimal dissipation) by a combination of analytical and numerical techniques. Their behavior at large velocities differs qualitatively from that for higher dimensional systems. In our generic model, a dissipative fluid is maintained in a non-equilibrium steady state by a stochastic or deterministic driving force. The velocity distribution for stochastic driving is regular and for infinitesimal dissipation, has a stretched exponential tail, with an unusual stretching exponent bQE = 2b, twice as large as the standard one for the corresponding d-dimensional system at finite dissipation. For deterministic driving the behavior is more subtle and displays singularities, such as multi-peaked velocity distribution functions. We classify the corresponding velocity distributions according to the nature and scaling behavior of such singularities.

I. INTRODUCTION A. Background and outline

The model of inelastic hard spheres is one of the most simple frameworks to describe granular gases (see e.g. [1,[START_REF]Theory of Granular Gas Dynamics[END_REF][START_REF] Barrat | [END_REF] for reviews and further references). The contraction of phase space due to dissipative collisions leads to a nonequilibrium behavior that is markedly different from that of equilibrium systems (non-Gaussian velocity distributions, counter-intuitive hydrodynamics, breakdown of kinetic energy equipartition etc [1,[START_REF]Theory of Granular Gas Dynamics[END_REF][START_REF] Barrat | [END_REF]). In this paper, we study in detail the limit of quasi-elasticity [4,5,6,[START_REF] Ramirez | Granular Gases[END_REF] with particular emphasis on one-dimensional systems, that have already been the subject of some interest [START_REF] Mcnamara | [END_REF]9,10,11].

The kinetic description is provided by the nonlinear Boltzmann equation. As we are interested in the velocity statistics of dissipative gases, we will restrict ourselves to homogeneous and isotropic solutions. Any spatial dependence will therefore be discarded. The time evolution of the velocity distribution function F (v, t) is then governed by the following equation [12,13],

∂ t F (v) + F F = I(v|F ) ≡ n dw g ν 1 α ν+1 0 F (v * * )F (w * * ) -F (v)F (w) . (1)
Here F F represents the action of a driving mechanism, that injects energy into the system, and counterbalances the energy dissipated by inelastic collisions. Consequently, the system is expected to reach a non-equilibrium steadystate. In the equation above g = vw denotes the relative velocity of colliding particles with g = |g|, n (• • • ) = (1/Ω d ) dn(• • • ) is an angular average over the surface area Ω d = 2π d/2 /Γ( 1 2 d) of a d-dimensional unit sphere, and g ν models the collision frequency. Note that in one dimension the integral n is absent. We have absorbed constant factors in the time scale. Here (v * * , w * * ) denote the restituting velocities that yield (v, w) as post-collisional velocities, i.e.

v * * = v -1 2 (1 + α -1 0 )(g • n)n ; w * * = w + 1 2 (1 + α -1 0 )(g • n)n, (2)
where (unit) vector n is parallel to the line of centers of the colliding particles. Note that nn is replaced by 1 in one dimension. The direct collision law is obtained from (2) by interchanging pre-and post-collision velocities and by replacing α 0 → 1/α 0 where α 0 < 1 is the restitution coefficient. Each collision leads to an energy loss proportional to (1 -α 2 0 ). Elastic collisions therefore correspond to α 0 = 1. In this article, we will consider the source term in (1) to be of the form

F F = ∂ • (aF ) -D∂ 2 F = γ∂ • (vv θ F ) -D∂ 2 F, (3)
where a = vv θ is a negative friction force, ∂ ≡ ∂/∂v is the gradient in v-space, and γ and D are positive constants. Two situations will be addressed : (γ = 0, D > 0) or (γ > 0, D = 0). They correspond respectively to stochastic White Noise (WN), or to deterministic nonlinear Negative Friction (NF). While the WN driving mechanism has been extensively studied [6,15,16], the Negative Friction has been introduced more recently [13]. The continuous exponent θ ≥ 0 selectively controls the energy injection mechanism. Schematically, increasing the value of θ corresponds to injecting more energy in the large velocity tail of the distribution. However two special values have been studied in the past [12,13,14,16,17,18], i.e. (i) the Gaussian thermostat (θ = 1), which is equivalent to the homogeneous free cooling state, where the system is unforced and the possibility of spatial heterogeneities discarded (see e.g. [16]), and (ii) the case θ = 0, referred to as 'gravity thermostat' [16], or as 'negative solid friction' [13,14] . We emphasize that γ and D are irrelevant constants that can be eliminated (see below), whereas the exponents θ and ν are fundamental quantities for our purposes. As it appears in Eq. ( 1), ν governs the collision frequency of the system: ν = 1 corresponds to hard-sphere like dynamics and ν = 0 to the so-called Maxwell model [12,19,20,21,22]. Our collision kernel generalizes these two cases to a general class of repulsive power law potentials, where ν is related to the power law exponent and the dimensionality (see [13]).

Under the action of the driving term F , the solution of (1) evolves towards a non-equilibrium steady state. We will be interested in the properties of the corresponding velocity distribution F (v), in the limit where α 0 → 1 -. Since the limit α 0 → 1 turns out to be singular in one dimension, attention must be paid to the fact that the value α 0 = 1 (elastic interactions) has to be analyzed separately. Indeed, when α 0 = 1 in one dimension, the collision law (2) simply corresponds to an exchange of particle labels. So the initial velocity distribution does not evolve in time. On the other hand, the steady state velocity distribution at any α 0 < 1 is independent of the initial condition. In dimensions higher than 1, this property holds for all values of α 0 ≤ 1. In other words, the one dimensional situation with α 0 < 1 exhibits universal features, unlike its elastic counterpart. The quasi-elastic limit α 0 → 1 -is therefore peculiar since a point with no universal properties (α 0 = 1) is approached via a 'universal' route (α 0 < 1). The resulting behavior of F shows some surprising features, that may be considered as mathematical curiosities, but are analytically challenging. It also turns out that they are numerically difficult to study. The numerical study relies on the DSMC (Direct Simulation by Monte Carlo) algorithm [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF], which allows us to obtain an exact numerical solution of the Boltzmann equation. As α 0 approaches 1, the memory of the initial conditions lasts for longer and longer times so that the computer time needed to reach the non-equilibrium steady state increases and simulations become more and more time-consuming.

The behavior of our system is somewhat simpler in the case where energy is injected by a stochastic force (WN), and we start by analyzing this driving mechanism in section II. It will be shown that the regular high energy tail of

F (v) ∼ exp[-v b ],
which holds in any space dimension [12,13,14], is preempted by a quasi-elastic tail characterized by a different stretching exponent b QE = 2b. This is a signature of the non-commutativity of the limits: v → ∞ and α 0 → 1 -. The case of driving through negative friction will be addressed in detail in section III. As already observed for the homogeneous cooling state of inelastic hard rods [4,5,9,17,18], the velocity distribution becomes singular in the QE-limit, where it may approach a multi-peaked solution, and not a Gaussian. Starting from a small-inelasticity expansion for the collision operator I(v|F ) in (1), we characterize the scaling behavior. By a combination of analytical work and numerical evidence, we propose a classification of the different types of limiting velocity distributions, several of which correspond to new types of solutions of the nonlinear Boltzmann equation.

B. Preliminary remarks

We start by introducing some notations and summarizing a few results [13] that are relevant for our study. In the subsequent analysis, it is convenient to introduce the variables p = (1 + α 0 )/2, q = (1 -α 0 )/2 so that p + q = 1, and to measure the velocities in units of the r.m.s. velocity. We study steady states and introduce a rescaled velocity distribution f (c) such that F (v) = v -d 0 f (v/v 0 ) where v 0 is the r.m.s velocity, c = v/v 0 , and d is the number of spatial dimensions. By definition, dcf = 1 and the normalization chosen reads dcc 2 f = d/2. After inserting the scaling ansatz in Eq.( 1), we have shown in [13] that a stable steady state for WN driving is reached provided b W N = 1 + ν/2 > 0, and for NF driving provided b N F = ν + 1 -θ > 0. When b < 0, the non-equilibrium steady state is unstable, i.e. it is a repelling fixed point of the dynamics. Our analysis should therefore be limited to the cases ν > -2 for WN and to ν > θ -1 for NF.

We have shown in [13] that the quantity b introduced above not only separates stable from unstable situations, but also governs the high energy tail of f (c). In marginal cases where b vanishes, f has a power-law tail. The freely cooling (θ = 1) Maxwell model (ν = 0) provides an illustration that has been discussed in [20,21,22]. On the other hand, when b > 0, f has a stretched exponential tail so that ln f (c) ∝ -c b at large c. This result holds in any dimension. In d = 1, the corresponding tail may however be 'masked' when α 0 is close to unity, a phenomenon already observed for hard rods (ν = 1) in [17]: the behavior ln f (c) ∝ -c b holds for c > c * (α 0 ), where c * (α 0 ) is an α 0 -dependent threshold. In the QE-limit, the threshold value c * (α 0 ) → ∞. As a consequence, considering the limit of large c at any finite α 0 , the standard behavior with exponent b is observed. Alternatively, taking first the limit α 0 → 1 -at fixed c , new tails may appear. It is the purpose of the present paper to study their properties. In addition, in the case of NF driving, f (c) becomes singular in the quasi-elastic limit. Our goal will then be to understand the underlying scaling behavior and to propose a classification of the various limiting shapes for the velocity distribution.

II. WHITE NOISE (WN) DRIVING

Unless explicitly stated, we limit ourselves to d = 1. In the case of WN driving the integral equation ( 1) for the scaling form, f (c) = v 0 F (cv 0 ), becomes with the help of the relation

I(v|F ) = v ν-1 0 I(c|f ): I(c|f ) = -Dv -ν-1 0 f ′′ (c) = -1 2 pqκ ν f ′′ (c). (4)
The second equality has been obtained by applying dcc 2 (• • • ) to the first equality, see [13]), yielding

Dv -ν-1 0 = -1 2 c 2 I(c|f ) = 1 2 pq |c -c 1 | ν+2 ≡ 1 2 pqκ ν , (5)
where the double brackets denote an average with weight f (c)f (c 1 ). Similarly, simple brackets denote an average with weight f (c).

It is next convenient to treat the 1-D collision term,

I(c|f ) = dc 1 |c -c 1 | ν α -ν-1 0 f (c * * )f (c * * 1 ) -f (c)f (c 1 ) , (6)
by using the inverse transformation, c = qu + pc * * 1 and c 1 = pu + qc * * 1 , where u = c * * and |c -c 1 | = α 0 |c -u|/p. We obtain

I(c|f ) = du|c -u| ν p -ν-1 f (u)f c -qu p -f (u)f (c) . (7)
In the quasi-elastic limit q → 0 + , p → 1 -, we perform the small-q expansion, following Refs. [4,5,6],

1 p ν+1 f c -qu p = (1 + (ν + 1)q)f (c) + q(c -u)f ′ (c) + O(q 2 ). (8)
Then we find to O(q) included,

I(c|f ) = q duf (u)|c -u| ν [(c -u)f ′ (c) + (ν + 1)f (c)] = q d dc f (c) du|c -u| ν (c -u)f (u). (9)
Note that these results hold irrespective of the driving mechanism. The second equality can be verified by evaluating the derivative. Inserting of ( 9) into (4) allows us to integrate (4) once, yielding:

f ′ (c) + f (c)(2/κ ν ) du|c -u| ν (c -u)f (u) = 0. ( 10
)
This equation can be integrated once more to obtain the implicit equation,

f (c) = C exp - 2 (ν + 2)κ ν du|c -u| ν+2 f (u) ∼ C exp - 2c ν+2 (ν + 2)κ ν (c → ∞), (11)
where C is an integration constant. The large velocity tail immediately follows. We recover a result already obtained in [6,17] for hard rods (ν = 1), as well as one for Maxwell models (ν = 0) in [18], and we note that in the QE-limit the exponent b QE = 2b W N = ν + 2. On the other hand, for any finite ε = 1 -α 0 , the large c-behavior is given by ln f (c) ∝ -c b with b = 1 + ν/2 [13]. These two facts show that the limits c → ∞ and ε → 0 do not commute,

f (ε, c) ∼ exp(-A c 1+ν/2 ) (standard tail: large c, fixed ε = 0) f (ε, c) ∼ exp(-A c ν+2 ), (QE-tail: small ε, fixed c). (12)
or, in a more rigorous formulation

lim ε→0 + lim c→∞ ln f (ε, c) c 1+ν/2 = C (standard tail) lim c→∞ lim ε→0 + ln f (ε, c) c ν+2 = C ′ (QE-tail). (13)
.

We have successfully tested these predictions against Direct Simulation Monte Carlo data [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF]. Figure 1 shows that the standard tail with exponent b QE = 1 + ν/2 is clearly observed for α 0 = 0 (see inset), while the QE-tail with exponent ν + 2 applies for α 0 = 0.995 (see main frame). Figure 2 conveys a similar message, and shows further that in two dimensions, Gaussian behavior is recovered, as expected, when α 0 → 1 (see the inset of the right hand side figure). This illustrates the qualitatively different nature of the quasi-elastic limit in d = 1, as opposed to higher dimensions. 1) with WN driving by the DSMC technique [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] for the strongly interacting so-called 'very hard particles' model (ν = 2) [19]). Two extreme cases have been simulated, one close to perfect elasticity (α0 = 0.995) where the QE-tail ln f vs. c ν+2 (main frame) is visible, and the other at complete inelasticity (α0 = 0), where the standard tail ln f vs. c 1+ν/2 (inset) is visible. ) with ln f (c) vs. c ν+2 (main frame) and vs. c 1+ν/2 (inset). In the main frame the dashed line is a guide for the eye corresponding to the expected behavior (12). Right: d = 2, ln f (c) vs c 2 (main) and vs. c 1+ν/2 (inset).

III. NEGATIVE FRICTION (NF) DRIVING

The scaling equation for the analog of (4) with NF driving becomes,

I(c|f ) = γ v ν+1-θ 0 d dc c|c| θ-1 f = pqκ ν 2µ θ+1 d dc c|c| θ-1 f . (14)
Here I(c|f ) takes the form (9), and the analog of ( 5) has been used to eliminate γ. Moreover, µ θ+1 = |c| θ+1 and κ ν = |c -c 1 | ν+2 . We therefore have to O(q) included,

f (c) dc 1 |c -c 1 | ν (c -c 1 )f (c 1 ) = κ ν 2µ θ+1 c|c| θ-1 f (c). ( 15
)
Previous studies of the free cooling regime of hard rods (θ = 1, ν = 1) have shown that f (c) becomes singular when α 0 → 1 -, and evolves into two symmetric Dirac peaks [4,5,9,17]. It is indeed easy to check that f (c) = [δ(c + a) + δ(c -a)]/2 is a solution of ( 15)), with a = 1/ √ 2 as required by our choice of normalization, c 2 = 1/2, and κ ν /(2µ θ+1 ) = 2 ν a ν+1-θ . Figure 3 shows that the approach to such a solution may be observed in the numerical simulations for (θ, ν) = (1, 1). In addition, one can check that f (c) = A[δ(c + a) + δ(c -a)] + Bδ(c) is also a solution of Eq. ( 15), provided that 2A + B = 1 and 4Aa 2 = 1 to enforce normalization. The DSMC results may indeed display such a three-peak structure (see Figure 4). Note that the parameters corresponding to Figures 3 and4 However, upon changing the parameters θ and ν, it appears that more complex shapes can be observed: f (c) may evolve towards a 4-, 5-, 6-peaks form, or other structures such as displayed in Figure 5 where f (c) seems to diverge at some points when α 0 → 1 -, with nevertheless a finite support. The diversity of the various velocity distributions obtained numerically calls for a rationalizing study. By a combination of analytical work and numerical evidence, we will propose below a classification of the different possible limiting velocity distributions. In addition, in the double peak case, two natural questions will be addressed: Are the peaks exemplified in Figure 3 self-similar ? If so, what is their shape ?

A. The double peak scenario : structure and scaling

We start by looking for scaling solutions to Eq. ( 1), and restrict our analysis to the limiting form with the symmetric double peak. As in [17] we take f (c) of the doubly peaked form,

f (c) = b 4E ψ 1 + bc 2E + ψ 1 -bc 2E , (16)
where the width E = q ω is expected to vanish when q → 0 + and b, ω, together with the function ψ are unknowns. We impose 1 ψ ≡ ψ = 1, and we choose x ψ = 0, which together with the condition c 2 = 1/2, implies

2 = 2(1 + 4E 2 x 2 ψ )
, where normalization requires that b → ( √ 2) -as q → 0 + . We note that asymmetric forms with ψ(x) = ψ(-x) may be realized (see [17] and later sections). The ansatz (16) allows us to resolve the structure of the Dirac peaks, shown in Figure 3, and to identify the type of self-similar behavior involved. However to this end, we need to expand the collision operator I(c|f ) to second order in the inelasticity ε = 1 -α 0 = 2q. Restriction to first order, as done in (9), enables us to show that the double Dirac form is a solution of the Boltzmann equation, but does not allow us to impose constraints on the shape of ψ and on the scaling exponent ω. Technical details can be found in the appendix, where it is shown that the equation fulfilled by ψ(x) reads,

qψ ′ (x) + 2E 2 (ν + 1 -2θ)xψ(x) + 2E ν+2 ψ(x) |x -y| ν (x -y) ψ -E 3 ν(ν + 1)ψ(x)(x 2 + y 2 ψ ) + 2E 3 [(ν + 1)(ν + 2) -2θ(θ + 1)] y 2 ψ ψ(x) +4E 3 θ(θ -1)x 2 ψ(x) + 2E ν+3 ψ(x) |y -z| ν+2 + O(Eqψ) = 0, ( 17
)
where y, z are dummy variables, and terms of O(Eqψ) have been neglected. This relation involves terms of various orders in inelasticity. Given that E = q ω , these terms are O(q a ) with a = 1, 2ω, (2+ν)ω, 3ω, (3+ν)ω, ω +1. Depending on the values of the parameters θ and ν one has to distinguish various possibilities to characterize the phase diagram, i.e. the physically allowed region of the (θ, ν)-parameter space. The stability criteria for the steady state (see [13]) constrain the phase diagram in Figure 6 to be inside the region [θ ≥ 0, ν ≥ θ -1].

1. Case ν > 0 (α-and β-scalings).

Type α-scaling: The terms of order q and q 2ω are the dominant ones in Eq. (17). So, ω = 1/2 and I. The stability of the non-equilibrium steady state requires that ν > θ -1, while on the diagonal ν = θ -1 the velocity distribution has a power-law tail [13]. Here α-, α ′ -and β-scalings are associated with a distribution with two Dirac peaks. For θ > 1 we observed numerically a solution with three Dirac peaks, while there does not seem to be a simple common feature for the distributions in the triangular "Zoo" region. An example of type α-scaling is given in Figure 7 (See also Figure 8 for type β, and Figure 9 for type α ′ ). Figures 12 and 13 together with Figure 5 above give an overview of several scaling shapes encountered in the Zoo region. The cross at (1,1) corresponds to the homogeneous cooling state of inelastic hard rods, explored in [17]. √ 2 corresponds to the peak of the distribution. Open symbols correspond to ν = 1.5 and filled symbols are for ν = 1.2 (same as in Figure 3). The inset shows the same results on a linear-log scale. In each case, the prediction of Eq. ( 19) for the scaling function is shown by the continuous curve. Note that no fitting parameter is involved.

ψ ′ (x) = -(ν + 1 -2θ)xψ(x). ( 18
)
The scaling function ψ is therefore Gaussian,

ψ(x) ∝ exp -(ν + 1 -2θ)x 2 . ( 19
)
Such a solution is meaningful only if ν + 1 -2θ > 0. This scaling behavior, hereafter referred to as type α (not to be confused with the restitution coefficient α 0 ) is compared in Figure 7 with Monte Carlo results. The agreement between the analytical prediction and the numerical data involves two aspects : first, the exponent ω = 1/2 allows us to rescale all distributions onto a single master curve. Second, this curve is exactly of the form (19), where the prefactor hidden in the proportionality sign ∝ follows from normalization. Note that the excellent agreement between numerical data and analytical prediction is therefore obtained without any fitting parameter.

Type β-scaling: On the line ν + 1 -2θ = 0, the term of order q 2ω in (17) vanishes. If 0 < ν < 1, the terms q and q (ν+2)ω can be balanced with the result ω = 1/(ν + 2), and one finds at large |x|, 20) where the prefactor is determined from the constraint dcψ = 1.

ψ(x) ∝ exp - 2 ν + 2 |x| ν+2 . ( 20
)
Unlike in α-scaling , it is not possible here to obtain ψ in close form. We will refer to (20) together with ω = 1/(ν + 2) as a scaling of type β. Figure 8 shows that this behavior is in good agreement with the simulation results. Note that Eq. ( 20) seems to hold also for small values of x, whereas it is a priori only valid to describe the large-x tail of ψ. We also note that the large-x tail (20) exhibits the same exponent ν + 2 as in White Noise driving (see Eq. ( 11)). The important qualitative difference between the velocity distributions reported in section II and here is: with WN driving f (c) is regular when α 0 → 1 -, and with Negative Friction it develops singularities.

On the line ν + 1 -2θ = 0, now with ν > 1, one can only balance the terms in q and q 3ω in ( 17). This leads to ω = 1/3, but the associated function ψ diverges for large arguments, and is thus unphysical. This is an indication that the double peak limiting form cannot be valid on the line ν + 1 -2θ = 0 if ν > 1. Monte Carlo results confirm this. They display a limiting form with three peaks in this region of parameter space (see Figure 4 and section III B for a more thorough discussion, in particular Figure 6).

The special point on that line with (θ, ν) = (1, 1) represents free cooling of inelastic hard rods, which has been studied in [17]. There it was shown that ( 16) holds, with ω = 1/3 and an asymmetric scaling function, behaving at large arguments as,

ψ(x) ≃ C exp 1 3 x 3 + o(1) (x → -∞) ψ(x) ≃ C ′ exp -2x y 2 ψ + o(1) with C ′ = C exp -1 3 y 3 ψ (x → +∞). ( 21
)
This asymmetry looks quite singular since most other scaling functions identified so far are symmetric. However, more exceptional cases with asymmetric scaling forms ψ can be found in Figure 10 (which corresponds to the point (0, -1), a case of marginal stability where ν = θ -1), as well as in the scaling shapes of the "Zoo" region of Figure 6. We also emphasize that the Dirac peaks appearing at the level of description when α 0 → 1 -are not artifacts of discarding any spatial dependence in (1), but provide the exact solution of the Boltzmann equation where due account is taken of the spatial degree of freedom of the particles. This has been confirmed in [5] and [17], where the velocity distributions of the homogeneous Boltzmann equation has been compared with its exact counterpart, obtained by Molecular Dynamics simulations.

2. Case ν = 0 (α ′ -scaling).

Type α ′ -scaling: When ν vanishes (Maxwell models), the first three terms on the rhs of ( 17) are of the same order, so that ω = 1/2 and ψ satisfies,

ψ ′ (x) + 2(1 -2θ)xψ(x) + 2ψ(x) x -y ψ = ψ ′ (x) + 4(1 -θ)xψ(x) = 0, (22)
since y ψ = 0. Its solution is,

ψ(x) ∝ exp[-2(1 -θ)x 2 ]. (23)
This scaling, coined α ′ , a priori holds for 0 ≤ θ ≤ 1, as follows from ν = 0 and the stability requirement ν + 1 -θ ≥ 0. However, when θ = 1 (free cooling), [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] becomes unphysical. This is a consequence of the peculiar behavior of the freely cooling one-dimensional Maxwell model: the velocity distribution, which is algebraic, does not depend on α 0 for α 0 < 1 [20,21]. In the (trivial) quasi-elastic limit, f can consequently not develop a singularity of any kind. The scaling ansatz ( 16) has to break down for θ = 1, and it does. For θ < 1, the simulation data are in good agreement with α ′ -scaling predictions (see Figure 9), where again no fitting parameter has been used.

3. Case ν < 0 (β-scaling).

Type β-scaling: Finally we investigate the region ν < 0, which is further confined by the stability requirements for the steady state, (θ ≥ 0, ν ≥ θ -1). We then have (2 + ν)ω < 2ω and the terms of order q and q (2+ν)ω balance each other in Eq. ( 17). This implies ω = 1/(ν + 2) and

ψ ′ (x) ψ(x) = -2 |x -y| ν (x -y) ψ . (24)
We recover the β-scaling, and in particular the large-|x| expression,

ψ(x) ∝ exp - 2 ν + 2 |x| ν+2 . (25)
In the region ν < 0, we have successfully tested the validity of β-scaling against Monte Carlo simulations. In some cases however, we note that the best rescaling with respect to inelasticity is obtained with an exponent ω opt that slightly differs from the predicted 1/(ν + 2). For instance, with (θ, ν) = (0.7, -0.2) we find ω opt ≃ 0.58 while 1/(ν + 2) ≃ 0.55. This could indicate that the scaling limit has not yet been reached, or it reflects the fact that for negative values of ν, it is more difficult to reach the steady state in Monte Carlo simulations. Here the collision frequency is dominated by encounters with |c 1 -c 2 | ≪ 1, that lead to a negligible change in the velocities of colliding partners. Consequently the numerical efficiency of our algorithm drops significantly.

B. Range of validity of scaling predictions: towards a phase diagram.

We have reported above a good agreement between DSMC calculations and the scaling predictions assuming the limiting double peak forms of α-,β-and α ′ -scaling, when ε = 1 -α 0 → 0 + , for several points in the (θ, ν)-plane. We have also shown that in some (complementary) regions of this plane the scaling hypothesis (16) does not provide an attracting fixed point solution of the stationary nonlinear Boltzmann equation (1) in the quasi-elastic limit. In fact, our analysis shows that physical solutions with α-scaling do not exist for ν ≥ 0, and ν + 1 < 2θ (e.g. (θ, ν) = (1.1, 0.3); (1.1, 1.1)), and likewise for β-scaling with ν > 1 and ν + 1 = 2θ (e.g. (θ, ν) = (1.1, 1.2)). In these regions we have no predictions.

Furthermore, in the triangular region [θ > 1, ν > 1, ν + 1 > 2θ] the NF driven kinetic equation admits -at least on the basis of the criteria developed -QE-limiting solutions with two peaks, consistent at least to second order in ε. The dynamics selects a different solution with three peaks (see Figure 4 with (θ, ν) = (1.1, 1.3)).

A systematic Monte Carlo investigation of various points (θ, ν) -shown as dots in Figure 6-reveals that the range of validity of α-scaling is in fact limited to θ ≤ 1 [24]. In principle it would be possible to repeat the analysis of section III A with a three-peak solution, however we did not try to carry out this analysis. The reason is three-fold: First, it would not explain why α-scaling is not selected by the dynamics in the angular region above. Second, it is cumbersome. Third, there is numerical evidence that f (c) may differ from the two-peak or three-peak form (see the Zoo region of Figure 6). Thus, such analysis would in any case not provide a complete picture. A numerical investigation appears unavoidable and was used to identify the different regions of the "phase diagram" shown in Figure 6. We summarize our main findings:

• α-Scaling holds for ν > 0 and ν > 2θ -1 as found analytically, with the restriction θ ≤ 1 that follows from numerical evidence.

• β-Scaling applies to the line 0 < ν = 2θ -1 < 1 and also to the region ν < 0, where the additional restriction ν > θ -1 follows from the stability requirement of the steady-state solution of Eq. ( 1). Figure 10 with (θ, ν) = (0, -1) shows a case of marginal stability where ν = θ -1. It is observed that the scaling exponent is still given by ω = 1/(ν + 2) but the form (25) breaks down (ψ becomes asymmetric).

• α ′ -Scaling is valid on the "Maxwell" line ν = 0 for θ < 1.

• Hard rods under free cooling, i.e. (θ, ν) = (1, 1), display specific scaling (see Eq. ( 21), Table I and Ref. [17] for details).

• None of the above scalings hold for θ > 1, where we have always observed a triple peak as in Figure 4. Figure 11 (with the same parameters as Figure 4) shows that the distributions are also self-similar, with an exponent ω = 1/2. Although, we have no prediction for the three-peak forms, we note that the scaling function in Figure 11 is compatible with Eq. ( 19), pertaining to α-scaling. This could be specific to the parameters chosen, since those value of θ and ν obey the inequalities, ν > 0 and ν > 2θ -1, where α-scaling provides a two-peak solution of the Boltzmann equation. Two-and three-peak shapes therefore seem to have common features. We did not explore self-similarity further in the three-peak region.

• There exists another triangular region in the (θ, ν)-plane (the Zoo in Figure 6), where f (c) does not evolve toward a two-peak or a three-peak form. In some instances, the Monte Carlo data are compatible with a fourpeak limit as ε → 0 + (see Figure 12 where only the sector with c > 0 has been shown). In some other cases, we observe precursors of what seems to be a six-peak form (see Figure 13). For both Figures 12 and13, it is difficult to decide if the limiting form for ε → 0 + will be a collection of Dirac distributions (with therefore a support of vanishing measure), or a distribution with finite support. We could however identify some points in the triangle where the limiting f (c) clearly is of finite support (see Figure 5).

To summarize, α ′ -and β-scaling apply in the whole domain where they provide a solution to the Boltzmann equation, but α-scaling has a restricted domain of relevance compared to the region where the corresponding solution is selfconsistent. The key features of the analytical predictions are recalled in Table I.

Scaling type rescaling exponent rescaling function

α ω = 1/2 ψ(x) ∝ exp(-(ν + 1 -2θ)x 2 ) α ′ ω = 1/2 ψ(x) ∝ exp(-2(1 -θ)x 2 ) β ω = 1/(ν + 2) ψ(x) ∼ exp(-2|x| ν+2 /(ν + 2)) at large |x| θ = 1, ν = 1 ω = 1/3
Eq. ( 21), ψ asymmetric TABLE I: Theoretical predictions for the different scaling behaviors in the two-peak region.

IV. CONCLUSION

We have studied the one-dimensional nonlinear Boltzmann equation in the limit of quasi-elastic collisions for a class of dissipative fluids where material properties are encoded in an exponent ν such that the collision frequency between two particles with velocities c 1 and c 2 scales like |c 1 -c 2 | ν . Two driving mechanisms have been considered: stochastic white noise (in which case the generic effects only depend on ν) and deterministic negative friction (in which case, in addition to ν, a second important exponent θ ≥ 0, characterizing the driving, has been considered). In both stochastic and deterministic cases, the quasi-elastic limit does not commute with the limit of large velocities. This is specific to one space dimension. There are however important differences between the two driving mechanisms. In the white noise case, the normalized velocity distribution f (c), suitably rescaled to have fixed variance, is regular and displays stretched exponential QE-tails of the form exp[-c ν+2 ]. On the other hand, f (c) with deterministic driving -which encompasses the much studied homogeneous cooling regime-develops singularities as ε ≡ 1 -α 0 → 0 + . The corresponding scaling behavior is particularly rich. We have classified the scaling forms encountered in several families, see Figure 6 for a global picture. Some regions of this (θ, ν)-diagram are well understood, such as regions α, α ′ and β. Some other domains resist theoretical understanding. Even if some progress might be possible in the three-peak region (one at c = 0, the two others at ±c * ), the situation in the central Zoo region of Figure 6 seems more difficult to rationalize, and computationally elusive, since one needs to reach extremely small values of ε to see the precursors of presumed singularities. Inserting the ansatz ( 16) into (A2), the term I (1) of order q reads (A4) The second order term I (2) in (A2) is

I (1) = -q b 4E
I (2) = q 2 2 b 4E 2 b 2E d dx

FIG. 1 :

 1 FIG.1: Rescaled velocity distribution f (c) obtained by solving numerically the 1-D Boltzmann equation(1) with WN driving by the DSMC technique[START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] for the strongly interacting so-called 'very hard particles' model (ν = 2)[19]). Two extreme cases have been simulated, one close to perfect elasticity (α0 = 0.995) where the QE-tail ln f vs. c ν+2 (main frame) is visible, and the other at complete inelasticity (α0 = 0), where the standard tail ln f vs. c 1+ν/2 (inset) is visible.

FIG. 2 :

 2 FIG. 2: Left: Velocity distribution obtained from DSMC simulations for weakly interacting 1-D particles (ν = -12 ) with ln f (c) vs. c ν+2 (main frame) and vs. c 1+ν/2 (inset). In the main frame the dashed line is a guide for the eye corresponding to the expected behavior(12). Right: d = 2, ln f (c) vs c 2 (main) and vs. c 1+ν/2 (inset).

  are quite close: (θ, ν) = (1.0, 1.2) and (1.1, 1.3) respectively.

FIG. 3 :

 3 FIG. 3: Velocity distribution obtained from DSMC simulations with NF driving for (θ, ν) = (1.0, 1.2) in 1-D. As ε = 1-α0 → 0 + , f (c) becomes increasingly peaked around c = ±c * ≡ ±1/ √ 2, and ultimately evolves into two symmetric Dirac distributions. All distributions displayed here and in other figures are such that dcf = 1, dccf = 0 and dcc 2 f = 1/2.

4 FIG. 4 :FIG. 5 :

 445 FIG.4: Same as Figure3for slightly different parameters (θ, ν) = (1.1, 1.3). The velocity distribution now reaches a three-peak structure when ε ≡ 1 -α0 → 0 + .

  b

FIG. 6 :

 6 FIG.6: Phase diagram in the quasi-elastic limit. Each dot corresponds to a set of numerical simulations at smaller and smaller values of ε = 1 -α0, performed to check the validity of the scaling behaviors summarized in TableI. The stability of the non-equilibrium steady state requires that ν > θ -1, while on the diagonal ν = θ -1 the velocity distribution has a power-law tail[13]. Here α-, α ′ -and β-scalings are associated with a distribution with two Dirac peaks. For θ > 1 we observed numerically a solution with three Dirac peaks, while there does not seem to be a simple common feature for the distributions in the triangular "Zoo" region. An example of type α-scaling is given in Figure7(See also Figure8for type β, and Figure9for type α ′ ). Figures12 and 13together with Figure5above give an overview of several scaling shapes encountered in the Zoo region. The cross at (1,1) corresponds to the homogeneous cooling state of inelastic hard rods, explored in[17].

1 FIG. 7 :

 17 FIG.7: Scaling behavior of type α. Plots of ε ω f (c) vs. x = (|c| -c * )/ε ω at θ = 1 for various inelasticities ε = 1 -α0 with ω = 1/2 as predicted. Here c * = 1/ √ 2 corresponds to the peak of the distribution. Open symbols correspond to ν = 1.5 and filled symbols are for ν = 1.2 (same as in Figure3). The inset shows the same results on a linear-log scale. In each case, the prediction of Eq. (19) for the scaling function is shown by the continuous curve. Note that no fitting parameter is involved.

PredictionFIG. 8 : 2 .

 82 FIG.8: Scaling of type β. Same plot as Figure7, displayingε ω f (c) vs. x = (|c| -c * )/ε ω at (θ, ν) = (3/4, 1/2) with ω = 1/(ν + 2) = 2/5 and c * = 1/ √ 2.The continuous curve shows the prediction of Eq. (20) where the prefactor is determined from the constraint dcψ = 1.

PredictionFIG. 9 : 2 .

 92 FIG. 9: Scaling of type α ′ . Same plot as Figure 7 and Figure 8, displaying ε ω f (c) vs. x = (|c| -c * )/ε ω at (θ, ν) = (0.2, 0.0) with ω = 1/2 and c * = 1/ √ 2. The continuous curve is for Eq. (23), i.e. ∝ exp[-(1 -θ)x 2 )].

FIG. 10 :

 10 FIG.10: Velocity distribution at (θ, ν) = (0, -1) for two inelasticities. f (c) develops a cusp at c = c * where c * is ε-dependent. Plotting ψ(x) = εf (c) vs. x = (c -c * )/ε (inset), shows that here the scaling form(16) applies with an asymmetric ψ, and ω = 1 = 1/(ν + 2). For such low values of ν, the CPU time required to gather statistical knowledge is particularly large.

FIG. 11 :

 11 FIG.11: Plots of ψ(x) = ε ω f (c) vs. x = (c -c * )/ε ω , with ω = 1/2 for the same parameters as in Figure4(θ, ν) = (1.1, 1.3), i.e. for f (c) approaching a structure composed by three Dirac peaks. Here, c * ≃ 0.76 denotes the position of the right peak seen in Figure4. The continuous curve shows expression(19).

6 FIG. 12 : 5 FIG. 13 : 2 d dc 2 f

 61251322 FIG. 12: Two Zoo members at (θ, ν) = (0.75, 0.3) (left) and (θ, ν) = (0.85, 0.5) (right). Note the x-and y-scale in the plot on the right. For these parameters one needs to decrease ε below 10 -6 to realize that a seemingly single peak is likely to split into two sub-peaks as ε → 0 + .

- 1 +

 1 ) dyψ(y)(1 -E(x + y)) dyψ(y)|x -y| ν (x -y) , (A3) and expanding (1 -E(x + y)) ν+1 further yields, xE(ν + 1) -ν(ν + 1) 2 E 2 dyψ(y)(x + y) 2 + E ν+1 dyψ(y)|x -y| ν (x -y) ψ(x) .

  dyψ(y)|x -y| ν+2 + 2 b ν+2 ψ(x) dyψ(y)|1 -E(x + y)| ν+2 (A5)

APPENDIX A

In this appendix, we expand the collision operator I(c|f ) defined in [START_REF] Ramirez | Granular Gases[END_REF] in powers of q = (1 -α)/2, up to second order. Such an expansion is required to unveil the internal structure of the singular peaks that develop as q → 0 + with driving by negative friction. Assuming that the functional form of the velocity distribution is given by ( 16), our goal is to obtain here the differential equation fulfilled by the scaling function ψ. Starting from

and we can keep only the largest term as q → 0 + , i.e.

Collecting terms yields finally,

The next step is to evaluate the right hand side of equation ( 14), by an expansion of the moments κ ν and µ θ+1 ,

With c close to -1/b, one can also perform the expansion

and the rhs of equation ( 16) can thus finally be written as

dydz|y -z| ν+2 ψ(y)ψ(z)] .

(A12)

We obtain (17), after simplification by the prefactors and integration with respect to x.


Mikko Ruohonen

Marko Mäkipää
email: marko.makipaa@uta.fi

Pekka Kamaja
email: pekka.kamaja@haaga-helia.fi

Competencies and Work Practices for Dynamic Distributed Software Development in Global Value Networks

Keywords: Software development, competencies, distributed work practices, global software engineering, dynamic offshoring

Offshoring of software development (SD) to cost competitive countries (CCC) has gained increased popularity in US and Western Europe since year 2000. Countries, such as India, have dominated the discussion but now it also seems that not just the labor costs matter. In the future, service levels, dynamic competencies building and community-based activities are also needed for managing dynamic distributed software development (DDSD) work. Instead of the one-way migration of RDI operations there are increasingly voices questioning the rationale of moving operations to CCCs. The key driver of this research-in-progress paper is in searching a new frameworks, tools and practices for managing DDSD work and developing SD operations evaluation solutions. The forthcoming practical outcomes described are both the improvements in SD work in the industry level and provision of enhancements for their current SD work performance assessment. Academic results will be discussed with European, US and Indian partner researchers in the context of changing dynamic sourcing i.e. onshore, nearshore, offshore activities in global value networks. The general objectives are to improve ICT-services companies' competencies and tools in i) assessing their SD work operations with more enhanced evaluation systems and ii) make visible practices in managing dynamic distributed sourcing network operations in global value networks.

Background and Rationale

Offshoring of software development to cost competitive countries has gained increased publicity in US and Western Europe since year 2000. India, China and even Latin America have been mentioned as potential cost competitive countries [START_REF] Ruohonen | Nearshoring or Offshoring -Comparing ITO and BPO practices between India and Europe[END_REF] [2] [START_REF] Adelakun | Demystification of Latin Americas Offshore Outsourcing Destination[END_REF]. Outsourcing of both information technology (ITO) and business and knowledge processes (BPO/KPO) has increased during the last two decades and now provides increasing business opportunities [START_REF] Lacity | Global information technology outsourcing: in search of business advantage[END_REF] [5] [START_REF] Saxena | Strategic Outsourcing of Engineering Processes Using Agile Methods[END_REF]. Cost competitive countries have dominated although it seems that not just the labor costs matter. In the future, service levels, dynamic competencies and community-based activities are also needed for managing dynamic distributed software development work. Cultural fit and suitability of administrative environment are important to notify when making outsourcing decisions [START_REF] Walsham | Foreword: special issue on information systems in developing countries[END_REF]. Outsourcing is too often considered one-sided: handing over assets, people, activities and knowledge to third-party management, but it can also be a contract for two-way collaboration to release your own knowledge and learning potential, while also releasing the provider's potential, for mutual gain [START_REF] Oshri | Managing Dispersed Expertise in IT Offshore Outsourcing: Lessons from Tata Consultancy Services[END_REF].

India has been a long time the giant of offshoring, but China has also grown. We must also remember that Latin America countries such as Brazil, Mexico and Chile and East European countries such as Poland, Czech Republic and Hungary are growing and competing in this arena. In addition on-shoring, i.e. keeping work inside your home country, is still a potential alternative for maintaining responsiveness, quality control and agility. This makes a challenging decision making arena for many software development companies. Strategic offshoring has created different business models which are in a state of flux. Organizations are looking for new managerial practices and comparisons of near-and offshoring models are frequently done.

Competencies for Distributed Dynamic Software Development

Strategy demands new competencies

The background of this paper is that software development companies are willing to assess new knowledge for arguing the research, development and innovation (RDI) operations in their home or foreign countries to make a successful combination of onshoring, near-shoring and off-shoring settings. Public discussion in many countries has been quite intensive before and after large outsourcing decisions. As it is noted, the low cost resources are not that clear when the impact of the hidden costs due to human resource management complexities or other structural and administrative issues related to business cultures in South and South-East Asian countries are considered.

Consequently the key driver of this research project is in searching new frameworks, tools and practices for managing dynamic distributed software development (DDSD) work and developing operations evaluation solutions. Therefore, the practical outcomes from this project are both the improvements in SD work management in the ICT industry level as well and provision of enhancements on their current performance management of software development. The overall goal encompasses the tools for management in adopting company management practices globally. Also, the optimization of the division of the RDI resources within the multitude of current and potential locations globally is considered.

Pure offshore outsourcing is changing to a more strategic direction which makes both on-shoring, near-shoring and offshoring a number of viable ways to implement software development. The process is bi-directional. For example, Indian IT-service giants have during the last years increasingly acquired shared services companies or invested in near-shoring operations in USA, Latin America and Europe.

India seems to be the most attractive country despite the promises of Chinese sourcing markets. Latin America and East European countries are joining the race. The fairly new EU member states are attractive for ICT services and also growing their ITO/BPO services exports. For the whole picture we should remember that the global sales of IT services is plus 1 trillion USD [START_REF] Gartner | IT Spending Forecast, Q4 2013 Update[END_REF] of which US-based companies such as IBM and Accenture normally take 60% market share. It is estimated [START_REF]The Hindu Business Line[END_REF] that total exports of Indian information technology enabled services (ITES) and business process outsourcing (BPO) business on financial year (FY) 2014 is on the level of 84-87 BUSD and the global market share approximately 4-5%. New EU countries are getting increasingly more visibility and growth.

Strategic offshoring has created different business models which are now in a state of flux. Network strategy requires development of value network to face growing and varying needs of customers. Indian companies attempt to move forward in the value chain in order to foster partnership and make customer innovations. Indian ICT service companies, previously known as "body shopping organizations" [START_REF] Heeks | India's Software Industry: State Policy, Liberalization and Industrial Development[END_REF], are now moving to countries with lower salary costs such as Vietnam, investing in Tier II cities such as Pune, Chandigarh, Jaipur in India to get lower costs of operations and better supply of manpower, set up excellence centers in China for using huge potential of Chinese engineering workforce and acquire IT companies from Europe and Latin America to get closer to customers. Indian companies are really networking to be nearer the customer but in the same time making effectiveness in their engineering work development. In addition to that Indian ICT service companies are heading for building vertical-specific service expertise in many industries and upgrading their knowledge in so-called business transformation outsourcing. Traditionally customer intimacy has been the competitive weapon for US and European ICT service companies. This strategy is based on middle-man model in which companies which are culturally nearer to customer can also use the costeffective offshore sourcing. The third model is to set up your own captive development center i.e. to execute internal offshoring.

Competencies for advanced performance management

Combining efficiency, productivity and effectiveness assessment makes an integrated framework. The studies of IBM [START_REF] Reddy | Software development and delivery performance measurement and management: Optimizing business value in software[END_REF] show interesting rationale for the project's performance management research. With eight selected topics researchers have found a clear link between business performance and software development work evaluation. Some challenges in measuring and managing software projects arise due to separate teams focused on development, build, testing, and deployment each having stove-piped processes. This leads to lack of timely information and in-context, objective, and honest assessment and insight into the status of software delivery projects. These challenges result even if all of the project members are co-located and working on a homogeneous environment.

The picture of software development gets more complicated when we add three additional dimensions commonly seen with many software delivery organizations:

-geographical/regional distribution of team members which adds poor communication, language, culture, time challenges and process gaps resulting in reworking,

-crossing organizational boundaries which leads to lack of effective collaboration, weak project governance, lack of domain expertise, poor line of business oversight, security of IP when outsourcing and -multiple team and heterogeneous infrastructure which adds more challenges to incompatible tools and repositories, unreliable access artifacts, lengthy onboarding and inflexible tooling integration. The dynamic distributed sourcing network is also evolving continuously. Therefore, we need managerial decision-making for finding the most successful teams, effective locations to source, combination of talents, balanced levels and a follow-up of product/service life-cycles.

Multicultural competencies for global software engineering

In addition to pure labor costs knowledge management, customer learning, managing different modes and lifecycles affect the final outcomes of dynamic distributed sourcing of software development. Multicultural project teams have a higher potential for success than single-culture teams do, but they also have a higher potential for failure. Even highly data-driven projects need to be carefully managed across cultures. We say: "it's the people, not the technologies, that spell or dispel success". Cultural differences in project management can be difficult to navigate, but whether or not you agree with the benefits of globalization, its effects will be felt for a long time to come, especially in the engineering industry. Taking the time to understand how culture affects a project and an organization not only makes smart business sense but also makes our everyday work life smoother and easier--the improved flow is priceless [START_REF] Hudson | The human touch: cohesive cross-cultural teams begin with savvy relationship management[END_REF].

It is vital to acknowledge the importance of cultural competence in order to act effectively and successfully in foreign cultures or in multicultural organizations. The concept of cultural competence is divided in knowledge, mindfulness and behavioral skills [START_REF] Thomas | Cultural intelligence: People skills for global business[END_REF]. Knowledge comprises information about the concept of culture, the ways culture affects behavior and different cultures. Mindfulness means mediating between knowledge and behavior. It means, for example, paying attention to our own assumptions of different cultures, breaking free of stereotypes and readiness to adjust our opinions of others. Acquiring behavioral skills means extending the set of possible behavioral ways and knowing in which situation and in which culture to use each one. Increasing cultural competence is a continuous process that can take considerable time. A good starting point for increasing cultural competence is offered by different cultural typologies.

One of the most widely accepted cultural typologies is presented by Hofstede [START_REF] Hofstede | Culture´s consequences[END_REF] [16] [START_REF] Hofstede | Culture's consequence: International differences in work related values[END_REF]. He distinguished cultures based on the differences in what they value and found five dimensions: power distance, uncertainty avoidance, individualismcollectivism, masculinity-femininity and long-term orientation. Trompenaars and Hampden-Turner [START_REF] Trompenaars | Riding the Waves of Culture: Understanding Cultural Diversity in Global Business[END_REF] have identified seven dimensions by which cultures can be differentiated. Five of these dimensions depict relationships among people: universalism-particularism, individualism-collectivism, specific versus diffuse, neutral versus affective and achievement versus ascription. The other two dimensions are time perspective and relationship with the environment. House et al. [START_REF] House | Culture, leadership, and organizations: The GLOBE study of 62 societies[END_REF] have presented nine dimensions in the GLOBE study of 62 societies, which help to understand the influence of cultural differences on leadership of organizations. These are power distance, uncertainty avoidance, human orientation, institutional collectivism, in-group collectivism, assertiveness, gender egalitarianism, future orientation and performance orientation.

The advantage of these models lies in their power to make sense of different cultures even if one does not have first-hand experience of a specific culture. However, the risk arises, that the models tend to simplify different cultures too much. For example, when we consider a country like India with many languages, castes and different living environments, it becomes evident that there is not a single homogenous Indian culture. According Jacob [START_REF] Jacob | Cross-cultural investigations: emerging concepts[END_REF] most countries are culturally heterogeneous consisting of several sub-cultures, which diminishes the value of cultural typologies describing whole nations. Instead of force-fitting countries in different classes, Jacob suggests the concept of crossvergence, which means "fusing together management practices of two or more cultures, so that a practice relevant to a heterogeneous culture can be assembled". As Jacob (ibid) states, the success of crosscultural managers is not defined by their knowledge of different culture typologies or ability to classify cultures, but more in their ability to find out what kind of leadership behavior best fits the culture and their ability to develop their own managerial skills according to that. More generally speaking, Fontaine [START_REF] Fontaine | Cross-cultural management: six perspectives[END_REF] gives two general focus areas for management in terms of cross-cultural management: understanding the past of the culture and shaping the future culture. As important as it is to understand the past of the culture, as we have seen in the number of studies concentrating on that, the more important it is to find ways to shape the future.

Research Settings: Questions, Approach, Results

The general objective of the project is to improve ICT-service (focused on software development ) companies' competencies and tools in i) assessing their software development work operations with more enhanced evaluation systems and ii) make visible work practices in managing dynamic distributed sourcing network operations in global value networks. This will improve ICT software sector to understand the changing sourcing environment of software development and service, detect their competencies in executing this work in different project and work settings and enable dynamic sourcing with multiple sites. The forthcoming results would be:

1. integrated framework for evaluation software development operations efficiency, productivity and effectiveness indicators (financial, quality and organizational measures) 2. creation of managerial practices for dynamic distributed software development sourcing networks and 3. comparison guidelines for evaluating multiple sourcing sites and locations both in Finnish/European and international offshore settings. The final goal is that the Finnish/European ICT and software industry can use these evaluation, work alignment and managerial practices for more agile and effective sourcing location management. This will make sourcing decisions more visible and easier to implement and provide agility for operations. Our research questions are:

1. How to find meaningful performance areas and create usable measures beyond traditional financial measures? 2. What are the topical areas for integrated evaluation framework and multisite sourcing management? 3. What are the competencies needed for creating and using implementable managerial work practices for multisite/location management? Outsourcing research in the context of work organizational management is a research field which needs both knowledge from information systems/software development (IS/SD) research, organizational development and work process management research. Evaluation and performance management literature is also needed in this project.

Our methodological approach in this project will be design/action research with participating companies [22] [23]. In design research researchers and company representatives are both creating new artefacts and evaluating them. It involves interplay of theoretical backgrounds and empirical investigations to produce viable, practical outcomes. Our first company, Case C, is an international software and services company specialized in telecommunications industry. The headquarters is located in Finland and in addition to that it has offices in 16 other locations in Asia, Europe, Middle-East and Australia. Net sales of the C group was 82.7 MEUR and the personnel approx. 700 employees on Y2013.

Since 1986 case C has helped more than 290 service providers across 86 countries to meet over one billion subscribers' communications and infotainment needs. C's service fulfilment, mediation, charging and policy control, and predictive social analytics products with implementation and professional services enable service providers to automate customer interactions and other business decisions for creating revenue, reduce costs and lessen churn.

The starting point in the DD-SCALE project of C is the increased interest in improving the monitoring of the capabilities involved in the efficiency monitoring practices. Secondly, the company is seeking solution to monitor the site specific efficiency figures that would increase the capability of managing the distribution of C's global research, development and innovation (RDI) resources.

C's solutions are built within its global RDI network. Various metrics are collected from sites, teams and programs, which are shared in multiple dashboards. RDI productivity is an area, where only limited metrics are available. The case company aims to get more fact-based information and tools from the DD-SCALE project for monitoring and managing productivity of the teams. A key strength of this project is data normalization across multiple companies.

Minicase 2: Advanced, large scale SD work evaluation -Case S PLC

The second company, Case S, is a large telecom network software and technology company with the full year sales of S 11.2 BEUR on Y2013. The number of employees is plus 20 000. Company claims to be as one of the world's top three mobile network infrastructure vendors serving more than 90 of the world's 100 largest tele-operators.

Case S's research interest in DD-SCALE project is the development of RDI efficiency analytics. So far the company is using a key performance indicator (KPI) approaches in evaluation of its RDI efficiency and the more conventional financial reporting, too. Case S has not only offshored activities but is also on-shoring a n umber of IT development resources from local service companies.

However, the monitoring of less visible factors, which are more intellectual in nature, escapes beyond the current system efficacy. In addition to that, the multitude of the complexities outside the company's borders in S's ecosystem imply factors influencing on company performance but are not currently considered in the S's performance analytics. The more far-reaching vision of the results of the DD-SCALE project is improving the decision making methodology of the distribution management of new sites globally. The current methodology would benefit from the findings in improving the software design work performance metrics.

Minicase 3: Common innovation system boost -Case A PLC

The third organisation, Case A, is a part of a global leading company in power and automation technologies with high market positions in selected main business areas. A Group has more than 150 000 employees in about 100 countries with 29 BEUR in revenues on Y2012. A Group is organized in five global divisions. The participating partner, case A, is the Collaborative Production Management software business (CPM).

The main interest of CPM unit in the DD-SCALE project is the overall effectiveness of A's innovation system for CPM software business that needs to be improved to facilitate further business growth of the global business in high growth markets. This means in particular accelerating the idea-to-market deployment speed in the end-to-end process from innovation idea to its actual deployment in the marketplace. The overall innovation system covers multiple A business units addressing the various industrial markets and the common software platform technology development unit at the corporate level, which is fulfilling the common technology needs of all the target markets. The distributed organization and global nature of business make this challenge particularly interesting.

Minicase 4: Organisational development in a multisite environment -Case N

The fourth and last company, Case N, is 25 years old company, specialized in the B2B software business providing solutions for ship design and operations The company has approximately 175 employees and offices and/or representatives in 8 countries such as Japan, Korea, China, Singapore, India and Romania. More than 95 % of its 18 MEUR turnover comes from exports and their business operations are truly global. The company has grown steadily and is expanding both geographically and in sales development. The RDI resources are located in three countries: Finland, India and Romania. Since March 2014 case N is owned by a Japanese company and taken out of the stock exchange.

Starting point for the DD-SCALE project in the case N is linked to the continuous development of organization. Creation of the most efficient organizational forms demands enhanced fact-based metrics. Especially, a more accurate and reliable performance reporting of different RDI sites and teams is needed.

Compared to the other business functions, software development is more collaborative in nature. This is seen especially in commonly used modern process frameworks for software engineering. Currently the most favored way to organize the software development is based on agile methods such as SCRUM.. Therefore, the focus of N in the DD-SCALE project is centered on the research of the collaboration and performance of software engineering teams in the multisite ITorganization such as case N. That means, for example, key performance indicators to monitor and determine the most efficient form of the organizations and distribution of work to maximize the overall performance of the company.

Conclusive Remarks

Our collaboration network of both research and educational institutes and global-scale software development organizations gives us an excellent viewpoint to both evaluate current situation and create new models and approaches for practice. Collaboration is ensured through active working with case companies and investigating on-going and background research. As it is noted by Prikladnicki and Audy [START_REF] Rikladnicki | Managing Global Software Engineering: A Comparative Analysis of Offshore Outsourcing and the Internal Offshoring of Software Development[END_REF] communications, methods, culture, and process details are just some of the facets of the unique characteristics of global software engineering environments. In the next ten years many of software development companies need to find their own competitive position in the evolving software markets, create a pattern of work practices capable for distribution of work, format new managerial practices for performance management in global value networks and promote innovations in open environments

Fig. 1 : 1 :

 11 Fig. 1: Information systems research in design research [21].

Acknowledgments. The authors would like to express their gratitude to TEKES -the Finnish Funding Agency for Technology and Innovation and participating companies for commitment and start of the DD-SCALE research project and fostering collaboration in the ICT industry.


Simone Opel

Axel Wellesen

Analysis of Real-Life Working Processes, Competencies and Operational Fields for the Usage in Vocational IT Education -Results of an Empirical Study Based on Job Offers

Keywords: Vocational IT Education, Vocational CS Education, Empirical Study, Learning Fields, Operational Fields, CS Working Areas, Working Skills, Working Processes

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Dual vocational education is a core element of the German educational system (2012: 551.272 new beginners in vocational education ([1], p. 7) vs. 495.088 first-yearstudents [3]). Dual vocational education consists of practical training at a company and theoretical education at school. In contrast to general secondary schools, the main intention of vocational schools is to teach relevant topics for the students' professional life to support development of working skills and professional action competency ( [START_REF]CMECA -Standing Conference of Ministers of Education and Cultural Affairs (Sekretariat der st¨andigen Konferen der Kultusminster der L¨ander in der Bundesrepublik Deutschland. Rahmenlehrplan f¨ur den Ausbildungsberuf Fachinfor-matiker/Fachinformatikerin (German)[END_REF], p. 14). Therefore, the curriculum also for the profession as computer specialist ("Fachinformatiker") is oriented towards the concept of learning fields.

Learning fields describe the competencies that students should achieve and specific content the students should know ( [START_REF]CMECA -Standing Conference of Ministers of Education and Cultural Affairs (Sekretariat der st¨andigen Konferen der Kultusminster der L¨ander in der Bundesrepublik Deutschland. Rahmenlehrplan f¨ur den Ausbildungsberuf Fachinfor-matiker/Fachinformatikerin (German)[END_REF], p. 10). For a better integration of theory and practice, each learning field has been based on characteristic real-life working processes. They are supposed to be put into practice by implementing "learning situations". Although this concept leaves room for creative implementation and therefore contains high potential for the development of varied activity-oriented learning situations, the concept is rarely put into practice.

To explore these reasons, Opel and Brinda [START_REF] Opel | Learning Fields in Vocational IT Education -Why Teachers Refrain From Taking an Opportunity[END_REF] [7] conducted an study with vocational school teachers. They could show that teachers in general are not familiar with the concept, but open-minded and motivated to implement it. However, they also revealed that "a significant part of the participants in the survey does not really know what it means to implement learning fields into learning situations."( [START_REF] Opel | Learning Fields in Vocational IT Education -How Teachers Interpret the Concept[END_REF], p. 156).The study also revealed that there are publications about the basics of the concept, but only very few about teaching with learning fields. In another interview study among IT and CS Training companies, Opel and Brinda [START_REF] Opel | Arguments for contextual teaching with learning fields in vocational it schools: Results of an interview study among it and cs training companies[END_REF] asked for typical working processes in the field of IT and CS. They came to the conclusion that most working processes cover main aspects of different learning fields, which corresponds to the results of the present work. However, the interviewed trainers mentioned that they would like to see more interdisciplinary projects, as well as activity-oriented and self-organised learning methods being put into practice at vocational schools. This could be achieved by consequently implementing the concept of learning fields.

To solve this problem, there is a larger project aiming at the development of a normative competence structure model, exemplary teaching material, tools and guidelines for creating learning situations using the learning field concept as well as a description of competencies the students should gain. Since learning fields are based on real-life working processes, this work contributes to these intentions by acquiring and analysing real-life operational fields, working processes and demanded competencies and how they are weighted in vocational reality.

Related Work

To enhance the acceptance of the curricular concept of learning fields, extended cooperation between companies and vocational schools could be useful. Repp et al. [START_REF] Repp | Lernortkooperation in der IT-Ausbildung -Kompetenzentwicklung in Projekten (German)[END_REF] discussed a project where apprentices as computer specialists for application development work on real projects from a nearby software company. After designing, realising and testing, the students present the project at the company and hand over the developed software. This way, a close connection with the vocational world can be achieved, which increases the motivation of the students. Furthermore, it offers the opportunity to integrate current technological and organisational developments from companies in vocational schools. Especially in the field of software development, there are chances for activity-oriented implementations of the according learning field by following professional software development process definition and thus represent real-life working processes. In contrast, there are no such implementations of learning fields concerning the field of IT Systems and networks. Instead, topics are treated in separate lessons and not in a wider context. This can cause problems when it comes to transfer the skills to vocational praxis [START_REF] Johlen | Lernfeld¨ubergreifender Zugang zu Betriebssysten und zur Netzwerktech-nik mit den Methoden der objektorientierten Programmirtechnik f¨ur die IT-Berufe (German)[END_REF].

Another study among over 600 companies and over 1000 apprentices [START_REF] Petersen | Die neuen IT-Berufe auf dem Pr¨ufstand. Erste Ergebnisse der bundesweiten IT-Studie (German)[END_REF] [10] explored IT working processes and education in business practice. The study revealed that lessons in vocational schools need to be improved because apprentices considered the demanded level to be too low. Furthermore, the apprentices criticised the lack of real-life relevance of the curriculum which could lead to a negative image of vocational schools. To improve this situation, the "SEDIKO" project aimed at designing learning fields as well as learning rooms [START_REF] Petersen | Neue Lernfeld-und Unterrichtsgestaltung in den IT-Berufen. Ergeb-nisse und Erkenntnisse aus dem Meodellversuche und Verbundprojekt SEDIKO (German)[END_REF]. After analysing and implementing the eleven learning fields, the conclusion was that these learning fields would only partially represent real-life working processes, since each single learning field only represents a part of a working process. The whole process can only be covered by their sum. This leads to the question whether there are learning fields that cover more of the demanded skills and competencies than others -and whether they should therefore be put into focus by vocational teachers.

All these studies confirm that the concept of learning fields seems to have a high potential which is not used consistently. A solution to this problem could be -besides a comprehensive description of competencies needed for the profession of computer specialist -the support of teachers by providing them exemplary material as well as guidelines for designing lessons based on the most demanded real-life working processes and skills.

Methodology

For a nationwide overview, we retrieved empirical data from 100 job offers listed in four of the biggest online job agencies: stepstone, monster, jobscout24 and the job market of the Federal Employment Agency (" obb¨orse der Bundesagentur f¨ur Arbeit"). Each job offer was analysed according to the demanded requirements and the described tasks. Both aspects were collected in two separate profiles. Requirements represent the competencies directly demanded by the companies. These include personal, operational and professional competencies as well as formal requirements and experience. Tasks represent the vocational operational fields and describe the tasks themselves as well as the area in which they take place (see fig. 1). Based on this data, a categorisation was developed by using an inductive approach. We defined a multi-level category system for requirements and tasks each. The resulting categories have been directly derived from job descriptions by using methods of content analysis and will be described in detail in the next section.

Results

To structure the data, we defined three generic terms in the task profile (Systems Administration, Software Engineering, Other) and five in the requirements profile (IT-technical area, Support and Documentation, Soft Skills and Other, Operational Processes and Structures, Formal Requirements). Each of them was further divided into several subcategories (see fig. 1).

Fig. 1 Example from the task profile

The core elements which structure the generic terms, are operations (e.g. installation, support) or competencies (e.g. the ability to communicate) and professional terms that give a more detailed description of the area in which the operations take place (e.g. installation of software/hardware).

The core elements which structure the generic terms, are operations (e.g. installation, support) or competencies (e.g. the ability to communicate) and professional terms that give a more detailed description of the area in which the operations take place (e.g. installation of software/hardware).

Thus, it is possible to reference the retrieved data with the curriculum and the goals and contents of the learning fields. Each task and each requirement was put into one of the categories by using keywords from their description. Categories that described similar things or could not clearly be divided have been summarised afterwards. At last, the resulting categories were assigned to the eleven learning fields in two steps. The first assignment was based on direct references, which means that the tasks and requirements described in the category had to be directly mentioned in the definition of the learning field. The second assignment was based on indirect references. These represent the previous knowledge that is additionally needed to successfully accomplish the required tasks. Figure 2 shows an excerpt of a matrix which contains the correlation between the three most demanded skills / working processes and the respective learning fields.

The largest number of mentioned tasks with a direct reference to the skills and contents were listed in the learning fields that cover IT Systems in general ("Maintenance of IT Systems", "Simple IT Systems" and "Networked IT Systems"). This emphasises the well-filled categories "user-support", "maintenance and updates", "fault analysis and correction"and "software configuration and administration". The most frequently demanded previous knowledge in the task profile originated from the learning field "Sources of information and working methods". Further learning fields that provide important previous knowledge for a successful accomplishment of the demanded tasks were "Professional English for technical occupations" and "Business processes and operational organisation".

According to the direct references to the skills and contents of the learning fields, the results for the requirements profile are similar. Differences appeared in terms of previous knowledge. Overall, most of the descriptions in the requirement profile were assigned to the learning field "Market and customer relationships", due to its relevance as previous knowledge. Furthermore, frequently demanded previous knowledge derived from the learning fields "Business processes and operational organisation", "Professional English for technical occupations" and "Sources of information and working methods".

The category "Soft skills and other" was evaluated separately, because its competencies are usually not domain-specific and thus cannot be assigned to a single learning field. The only exception is the subcategory "Knowledge of English", which can be clearly assigned to the learning field "Professional English for technical occupations". The largest number of mentions could be found in this category (demand in 56.0% of all job offers). Of similar importance are the demanded social skills "communication skills"(51.0% of all job offers) and "Ability to work in teams"(44.0% of all job offers).

Overall, the majority of working processes originate from the field of IT Systems in general, whereas the demanded previous knowledge occasionally varies, depending on the branch of business.

Conclusion

The question of this study was: What are the real-life working processes of computer specialists? The results show that the main topics all deal with IT systems in general. The most sought-after skills and competencies therefore derive from the learning fields "Networked IT Systems", "Simple IT systems" and "Maintenance of IT systems". The demanded tasks and skills involve installation, configuration, administration and support. The most sought-after previous knowledge can be assigned to the learning fields "Professional English for technical occupations", "Sources of information and working methods" and "Market and customer relationships". It could be valuable for vocational schools to focus on these learning fields in developing learning situations, since they represent the main aspects of the real-life working processes in the field of IT and CS. In addition, the development of social skills must be better taken into consideration across all learning fields.

The next steps will be to connect all data to a normative competence model for vocational computer science education as a general description of all competencies to be gained, furthermore the development of exemplary learning situations based on these results as well as suitable guidelines to support vocational teachers in developing their own lessons.

Fig. 2 .

 2 Fig. 2. Learning fields and assigned tasks and requirements