# Non Poissonian statistics in a low density fluid

Paolo Visco, Frédéric van Wijland, Emmanuel Trizac

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Paolo Visco, Frédéric van Wijland, Emmanuel Trizac. Non Poissonian statistics in a low density fluid. Journal of Physical Chemistry B, 2008, 112 (17), pp.5412. 10.1021/jp800333h. hal-00303732

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Non Poissonian statistics in a low density fluid

Paolo Visco,¹ Frédéric van Wijland,² and Emmanuel Trizac¹

¹Université Paris-Sud, LPTMS, UMR 8626, Orsay Cedex, F-91405 and CNRS, Orsay, F-91405

²Laboratoire Matière et Systèmes Complexes (CNRS UMR 7057), Université Denis Diderot (Paris VII), 10 rue Alice Domon et Léonie Duquet, 75205 Paris cedex 13, France

(Dated: November 19, 2018)

Our interest goes to the collisional statistics in an arbitrary interacting fluid. We show that even in the low density limit and contrary to naive expectation, the number of collisions experienced by a tagged particle in a given time does not obey Poisson law, and that conversely, the free flight time distribution is not a simple exponential. As an illustration, the hard sphere fluid case is worked out in detail. For this model, we quantify analytically those deviations and successfully compare our predictions against molecular dynamics simulations.

The purpose of the present paper is to address, for an arbitrary low density fluid in a stationary state, basic properties that have been under-appreciated or overlooked, which bear upon the collisional statistics: What is the probability distribution $P(\mathcal{N}, t)$ of the number of collisions $\mathcal{N}$ suffered in equilibrium by a tagged particle over a given duration $t$? Conversely, what is the probability distribution function of the free flight time, $P_{\text{FFT}}(\tau)$, again for a tagged particle? Such fundamental questions, relevant in their own right, have also consequences on the evaluation of transport coefficients and when transposed to the related realm of granular gases [1, 2], directly quantify dissipation. Our message is that intuitive expectation fails –except in highly untypical cases– as far as the collisional statistics is concerned, for an interacting fluid in and also, arguably less surprisingly, out of equilibrium. All results reported are new, together with the kinetic theory techniques used.

Whereas in a dense fluid, velocity correlations and hydrodynamic effects are responsible for a non trivial collisional statistics, one could naively expect that in the dilute limit where collisions become uncorrelated and molecular chaos is enforced [3, 4], collisional events define a Poisson process $P(\omega)$ so that $P(\mathcal{N}, t) = \exp(-\omega t)(\omega t)^{\mathcal{N}/\mathcal{N}^*}$, where $\omega$ is the mean collision frequency (i.e. $\langle \mathcal{N} \rangle/t \to \omega$ at long times, where the brackets denote an ensemble average). The corresponding free flight time distribution would then be

$$P_{\text{FFT}}(\tau) = -\frac{d}{d\tau} P(\mathcal{N} = 0, \tau) = \omega e^{-\omega\tau}. \quad (1)$$

However, as we shall see below –and this seems to have been ignored in the literature [5]– such a point of view is flawed. In essence, the collision frequency for a particle with velocity $\mathbf{v}$ depends on $\mathbf{v}$ (it generically increases with $v = |\mathbf{v}|$), which in turn induces correlations between successive collision times. After general considerations that encompass equilibrium and non equilibrium steady states, we will show that the collisional statistics is generically non Poissonian. This is what prompted us to focus on the simplest -analytically tractable- interactions and consider the equilibrium hard sphere fluid for illustrative purposes. Such a model is one of the most useful paradigms in statistical mechanics and has played an essential role in the development of the theory of liquids [6]. It is one of the simplest system exhibiting a phase transition. Remarkably, it does not only provide a valuable theoretical starting

point, but also enjoys direct experimental realizations [7]. It is therefore surprising that such a well studied system yields non trivial properties in a limit where little would have been expected. We will see that at late times, when $\omega t \gg \mathcal{N}$, the effect of such correlations is that $P(\mathcal{N}, t)$ is of Poissonian form but with a renormalized frequency $\omega/\sqrt{2}$ instead of $\omega$. This result holds irrespective of space dimension. In addition, explicit and accurate results will be reported for the number of collision cumulants $\langle \mathcal{N}^p \rangle_c$. Our analytical predictions will be compared to numerical simulations.

For the sake of simplicity, we begin the analysis by the free flight time distribution $P_{\text{FFT}}(\tau)$. The evolution of a tagged particle in a large stationary homogeneous fluid defines, in the low density limit, a Markov process where the transition rates can be computed from a linearized Boltzmann equation, see e.g [8]. In simplified situations, the velocity dependent collision rate $r(v)$ can be computed analytically; the inset of Fig. 1 shows an illustrative example for hard spheres in equilibrium. Velocities are expressed here in rescaled units, and for equilibrium situations, the velocity distribution function reads, with $d$ the space dimension

$$\phi(\mathbf{v}) = \frac{1}{\sqrt{2\pi}d} e^{-v^2/2}. \quad (2)$$

Out of equilibrium, $\phi$ is a stationary measure. In any case, the mean collision frequency $\omega$ follows from $\phi$:

$$\omega = \int \phi(\mathbf{v}) r(v) d\mathbf{v} = \langle r(v) \rangle. \quad (3)$$

From the Markovian property, it follows that the conditional probability of having a free flight time $\tau$ given a velocity $\mathbf{v}$ reads $P_{\text{FFT}}(\tau|\mathbf{v}) = r(v) \exp(-r(v)\tau)$. To proceed further and obtain $P_{\text{FFT}}(\tau)$ from some average of $P_{\text{FFT}}(\tau|\mathbf{v})$, attention must be paid to the fact that the relevant weight to use is not the velocity distribution $\phi(\mathbf{v})$ itself, but the velocity on collision, $r(v)\phi(\mathbf{v})/\omega$. The prefactor $r(v)$ biases the distribution toward more energetic events and accounts for the fact that in a given time interval, a particle with a larger than typical velocity will experience more collisions. We therefore obtain

$$P_{\text{FFT}}(\tau) = \int d\mathbf{v} \frac{r^2(v)}{\omega} e^{-r(v)\tau} \phi(\mathbf{v}). \quad (4)$$
This expression explicitly differs from the result reported in [9], where $r^2/\omega$ is replaced by $r$ (in other words, the weight used in [9] is $\phi$ and not $r\phi/\omega$). To see why such an approach is incorrect, one can compute the mean collision time $\langle\tau\rangle = \int \tau P_{\text{FFT}}(\tau) d\tau$, that should be equal to $1/\omega$. This is indeed the case with the distribution given in (4), whereas the formula of Ref. [9] gives $\langle\tau\rangle = \langle 1/r \rangle$, which differs from $1/\langle r \rangle = 1/\omega$ [10]. More importantly, upon neglecting the $v$ dependence of the rate $r$ (i.e. assuming $r = \omega$), the integral in (4) is readily integrated and yields expression (1) for $P_{\text{FFT}}$. It has been shown that a $v$-independent collision rate corresponds to particles interacting via an inverse power law pair potential with exponent $2d-2$ [4], which defines the so-called Maxwell model [11], a particularly convenient framework in kinetic theory. Maxwell molecules are nevertheless highly untypical and for any other fluid, $r$ depends on $v$ so that (4) cannot be exponential. We therefore conclude here that the collisional statistics is in general non Poissonian, except for Maxwell molecules where successive collisions turn out to be uncorrelated. We will clarify below the conditions for the occurrence of correlations, and show that while the $v$ dependence of $r$ is a necessary condition for non Poissonian behavior, it is in general not sufficient.

After the previous qualitative remarks, our goal is to quantify the deviations for Poissonian behavior, and to this end, we hereafter consider the prototypical hard sphere model where the frequency $r(v)$ takes the form [8]:

$$r(v) = \frac{\omega}{\sqrt{2}} \left( \frac{v^2}{d} {}_1F_1 \left( \frac{1}{2}, 1 + \frac{d}{2}, -\frac{v^2}{2} \right) + e^{-\frac{v^2}{2}} {}_1F_1 \left( \frac{d-1}{2}, \frac{d}{2}, \frac{v^2}{2} \right) \right), \quad (5)$$

where ${}_1F_1$ denotes a confluent hypergeometric function of the first kind. Although a closed-form expression cannot be obtained for $P_{\text{FFT}}(\tau)$ due to the lack of simplicity of the collision rate $r(v)$, finding the large $\tau$ behavior calls for a saddle point approximation for the integral appearing in Eq. (4), which yields

$$P_{\text{FFT}}(\tau) \stackrel{\omega\tau \gg 1}{\sim} \exp\left(-\frac{\omega\tau}{\sqrt{2}}\right) \frac{\omega}{2} \left(1 - \frac{2}{d} + \frac{\omega\tau}{\sqrt{2}d}\right)^{-d/2}. \quad (6)$$

Interestingly, to leading order, $\tau$ is distributed exponentially, as naively expected [see Eq. (1)], but with a renormalized rate $\omega/\sqrt{2}$. The validity of expression (6) is illustrated in Fig. 1, which displays results of numerical simulations. We have checked that the molecular dynamics data in Fig. 1 precisely coincide with the numerical integration of Eq. (4) for all velocities (not shown).

We also note here that very similar considerations hold for the distribution of path lengths: for a particle with velocity $v$, the free flight distance (FFD) $\ell$ travelled in a time $t$ is $\ell = vt$, so that $P_{\text{FFD}}(\ell|\mathbf{v}) = v^{-1}P_{\text{FFT}}(\ell v^{-1}|\mathbf{v})$. $P_{\text{FFD}}$ then follows from the counterpart of Eq. (4)

$$P_{\text{FFD}}(\ell) = \int d\mathbf{v} \frac{r^2(v)}{\omega v} \exp\left(-\frac{r(v)}{v}\ell\right) \phi(\mathbf{v}). \quad (7)$$

<figure><img src="image_2.png" /><figcaption>Figure 1 Free flight time distribution as a function of $\omega\tau$, on a logarith scale, for a two dimensional hard disc gas ($d=2$). The circles correspond to the results of molecular dynamics simulations at density $\rho = 0.04\sigma^{-2}$ where $\sigma$ is the disc's diameter, with $N=1000$ particles. The dotted line shows the prediction of Eq. (6) while Eq. (1) is shown with a dashed line. The inset shows the tagged collision frequency $r(v)/\omega$ versus $v$, also for $d=2$. The value $1/\sqrt{2}$ at $v=0$, indicated with an arrow, is $d$-independent.</figcaption></figure>

The resulting probability density is not a simple exponential, at variance with a claim sometimes found in the literature (see e.g. [12, 13]). However, a saddle point computation akin to that leading to (6) provides here the long distance behavior of $P_{\text{FFD}}$, which is governed by the minimum of the function $r(v)/v$ [14]. This leads a) to $P_{\text{FFD}}(\ell) \propto \exp(-\omega\langle v \rangle^{-1}\ell/\sqrt{2})$ for $\ell$ much larger than the mean free path, and b) to the remark that for the Maxwell model with a $v$ independent rate $r$, the minimum of $r(v)/v$ is reached for $v \to \infty$ and vanishes, which leads to a non-exponential large $\ell$ limiting behavior [15].

We now turn to the related but more complex question of the number of collisions. We introduce the joint probability $f(\mathbf{v}, \mathcal{N}, t)$ of having velocity $\mathbf{v}$ and having suffered $\mathcal{N}$ collisions in a time window $t$, for our tagged particle. The corresponding time evolution, again in the dilute limit, follows from the linear Boltzmann-like equation

$$\partial_t f(\mathbf{v}_1, \mathcal{N}, t) = \int d\mathbf{v}_2 \int d\hat{\sigma}(\mathbf{v}_{12} \cdot \hat{\sigma}) \theta(\mathbf{v}_{12} \cdot \hat{\sigma}) \\ \left[ f(\mathbf{v}_1^{**}, \mathcal{N}-1, t)\phi(\mathbf{v}_2^{**}) - f(\mathbf{v}_1, \mathcal{N}, t)\phi(\mathbf{v}_2) \right], \quad (8)$$

where $\theta$ is the Heaviside function, $\mathbf{v}_{12}$ is the relative velocity, $\hat{\sigma}$ is a unit vector and the ** superscript refers to pre-collisional velocities: $\mathbf{v}_1^{**} = \mathbf{v}_1 - (\mathbf{v}_{12} \cdot \hat{\sigma})\hat{\sigma}$ and $\mathbf{v}_2^{**} = \mathbf{v}_2 + (\mathbf{v}_{12} \cdot \hat{\sigma})\hat{\sigma}$. The latter equation encodes a full description of the collisional statistics for the tagged particle, in the low density limit. Here again we stress that such an analytical approach can be extended to other interaction potentials, leading to the already mentioned remark that, apart from Maxwell molecules, it does not admit a Poisson solution. Moreover, the above equation does not admit a stationary solution in the long time limit, due to the time dependent behavior of the collision number $\mathcal{N}$. More precisely, we expect the large time
dependence to be exponential, as a consequence of the linear character of the equation. For analytical progress, it turns convenient to introduce the generating function $\hat{f}$ through

$$ \hat{f}(\mathbf{v}, \lambda, t) = \sum_{\mathcal{N}=0}^{\infty} e^{-\lambda \mathcal{N}} f(\mathbf{v}, \mathcal{N}, t). \quad (9) $$

Of course, upon summing (8) over all possible values of $\mathcal{N}$, in the equilibrium state one recovers $\sum_{\mathcal{N}=0}^{\infty} f(\mathbf{v}, \mathcal{N}) = \hat{f}(\mathbf{v}, 0, t) = \phi(\mathbf{v})$, the Maxwell-Boltzmann distribution (2). It then appears that the cumulant generating function $\mu(\lambda)$, such that in the large time limit

$$ \langle \mathcal{N}^p \rangle_c \stackrel{\omega t \gg 1}{\sim} t(-1)^p \left. \frac{\partial^p \mu}{\partial \lambda^p} \right|_{\lambda=0} , \quad (10) $$

is the largest eigenvalue of an evolution operator that straightforwardly follows from (8) [15]. Furthermore, $\mu(\lambda)$ is directly related to the large time behavior of $P(\mathcal{N}, t)$ through its large deviation function $\pi$, defined as

$$ P(\mathcal{N}, t) \stackrel{\omega t \gg 1}{\sim} e^{t\pi(n)} \quad (11) $$

where $n = \mathcal{N}/t$. Indeed, $\pi$ is the Legendre transform of $\mu$:

$$ \pi(n) = \min_{\lambda} (\mu(\lambda) + \lambda n) . \quad (12) $$

The quantity $\mu(\lambda)$ therefore bears an important physical information, and has been the technical focus of our study.

From perturbation theory, we have obtained the behavior of $\mu$ at large $\lambda$ in the form

$$ \mu(\lambda) \sim \frac{\omega}{\sqrt{2}}(e^{-\lambda} - 1) + \mathcal{O}(e^{-2\lambda}) , \quad (13) $$

which implies that

$$ P(\mathcal{N}, t) \sim \frac{e^{-\frac{\omega t}{\sqrt{2}}}}{\mathcal{N}!} \left( \frac{\omega t}{\sqrt{2}} \right)^{\mathcal{N}} , \quad \text{for } \mathcal{N} \ll \omega t . \quad (14) $$

For this Poissonian behavior $P(\omega/\sqrt{2})$, the large deviation function easily follows:

$$ \pi(n) = n - n \log(n\sqrt{2}/\omega) - \omega/\sqrt{2} . \quad (15) $$

We note that Eq. (14) with $\mathcal{N} = 0$ is compatible with the time integral of the leading exponential order of free flight time distribution given in (6), as it should.

However, the dependence embodied in Eqs. (13) and (14) follows from a large $\lambda$ expansion and only holds for $\mathcal{N} \ll \omega t$ (hence $n/\omega \to 0$). It is therefore not indicative of the typical behavior, for which it is more useful to consider the low $\lambda$ limit. We then dwell on the remark made after (9) that for $\lambda = 0$, we have $\hat{f}(\mathbf{v}, 0, t) = \phi(\mathbf{v})$, which leads to the approximation scheme where $\hat{f}(\mathbf{v}, \lambda, t)$ is taken as a Gaussian with temperature $T(\lambda)$ and the requirement that $T(0) = 1$ [16]. This approximation is the lowest order of a more systematic expansion (see [15]), but it provides a useful and reasonably accurate information given its simplicity. The idea is

<figure><img src="image_3.png" /><figcaption>Figure 2 Large deviation function $\pi(n)$ characteristic of the long time behavior of the probability $P(\mathcal{N}, t)$ of suffering $\mathcal{N}$ collisions in a time $t$ (and defined by $\pi(\mathcal{N}/t) \sim t^{-1} \log P(\mathcal{N}, t)$ for $\omega t \gg 1$). The symbols correspond to molecular dynamics measures for a hard disc system with $N = 1000$ particles at two different but low densities. The solid line shows the Gaussian result following from Eqs. (16) while the dashed line is the large deviation function $\pi(n) = n - n \log(n/\omega) - \omega$ associated to the Poisson law $P(\omega)$. On the graph, the time scale has been set by the choice $\omega = 1$.</figcaption></figure>

to project the Boltzmann equation (8) governing the evolution of $\hat{f}(\mathbf{v}, \lambda, t)$ onto the first two velocity moments. This leads to

$$ \mu(\lambda) = - \frac{\omega}{\sqrt{2}} (1 - e^{-\lambda}) \sqrt{1 + \frac{T(\lambda)}{T_0}}, \quad (16a) $$

$$ \text{with } T(\lambda) = \frac{\sqrt{2}T_0}{\sqrt{1 + e^{\lambda}}}. \quad (16b) $$

It can be checked that $\mu(\infty) = -\omega/\sqrt{2}$, as implied by Eq. (13). The corresponding first three cumulants follow from (10) and read, irrespective of dimension $d$

$$ \frac{\langle \mathcal{N} \rangle_c}{\omega t} = 1 \quad (17a) $$

$$ \frac{\langle \mathcal{N}^2 \rangle_c}{\omega t} = \frac{9}{8} \quad (17b) $$

$$ \frac{\langle \mathcal{N}^3 \rangle_c}{\omega t} = \frac{289}{256} . \quad (17c) $$

These values are compared to molecular dynamics simulation data in Table I. Moreover, the large deviation function $\pi(n)$ of collisions, which follows from (12), appears to be in excellent agreement with its molecular dynamics counterpart, see Fig. 2.

At this point, it proves instructive to consider the Lorentz gas with only one mobile particle and a collection of spherical fixed scatterers. One readily gets $r(v) \propto v$, but the collisional statistics is nevertheless Poissonian. The reason is that the velocity modulus of the mobile particle is constant along the
<table><caption>TABLE I Cumulants for the number of collisions $\mathcal{N}$ from molecular dynamics simulations (performed on a two dimensional system with reduced density $\rho\sigma^2 = 0.04$), and comparison with both the Poisson $P(\omega)$ result and Gaussian approximation (17).</caption><thead><tr><th></th><th>⟨N⟩<sub>c</sub>/t</th><th>⟨N<sup>2</sup>⟩<sub>c</sub>/t</th><th>⟨N<sup>3</sup>⟩<sub>c</sub>/t</th></tr></thead><tbody><tr><td>ωt = 10</td><td>1.</td><td>1.123</td><td>1.129</td></tr><tr><td>ωt = 50</td><td>1.</td><td>1.135</td><td>1.105</td></tr><tr><td>Poisson</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Gaussian</td><td>1</td><td>1.125</td><td>1.129</td></tr></tbody></table>

trajectory: in other words, there is no thermalization. It therefore appears that non Poissonian behavior arises from two key properties, that induce collisional correlations: first, the collision rate depends on the velocities, and, second, the particle thermalizes to some stationary non singular measure. These criteria also apply out of equilibrium.

In conclusion, we have shown that the statistics of the number of collisions $\mathcal{N}$ experienced by a tagged particle in a low density homogeneous and stationary fluid, in or out of equilibrium, is a subtler quantity than it might seem. Our general statements have been illustrated with the hard sphere fluid. For the distributions of both $\mathcal{N}$ and the related free flight time –for which several incorrect results may be found in the literature–, we have quantified the corresponding non Poissonian behavior which follows from the simple physical ingredient that a particle with a high velocity statistically collides more often than a typical particle. A key quantity in the theoretical analysis is the cumulant generating function $\mu(\lambda)$, which can be computed explicitly for large $\lambda$ and approximately for small $\lambda$. The Gaussian ansatz worked out here can be considered as the lowest order of a systematic expansion. The resulting analytical predictions have been confronted against molecular dynamics numerical simulations, with a very good agreement. These numerical results show that the deviations from Poisson behavior $P(\omega)$, although not dramatic –which may be the reason why they are underdocumented in the literature– are nevertheless clearly observable. In particular, we have obtained the a priori surprising result that for long times, the distribution of $\mathcal{N}$ is Poissonian, but with a “dressed” rate $\omega/\sqrt{2}$. Conversely, the distribution of free flight time $\tau$ is exponential, with a behavior $\propto \exp(-\omega\tau/\sqrt{2})$.

Acknowledgements We would like to thank J. Piasecki,

J.M.J. van Leeuwen, M.H. Ernst, D. Frenkel and H. van Beijeren for useful discussions. This work was supported by the French Ministry of Education through a JCJC ANR grant.

## References and Notes

[1] N. V. Brilliantov and T. Poschel, Kinetic Theory of Granular Gases (Oxford University Press, Oxford, 2004).

[2] A. Barrat, E. Trizac and M.H. Ernst, J. Phys.: Condens. Matter, **17**, S2429, (2005).

[3] P. Résibois and M. de Leener, Classical Kinetic Theory of Fluids, John Wiley and Sons (1977).

[4] C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, 2000).

[5] The fact that $\mathcal{N}$ is not a Poisson variable has already been reported in the literature, see e.g. L. Lue, J. Chem. Phys. **122**, 044513 (2005) for numerical data on hard spheres, but to the best of our knowledge, no analytical results have been obtained.

[6] J. P. Hansen and I. R. MacDonald, Theory of Simple Liquids (3rd edition), Academic Press, 2006.

[7] P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, Phys. Rev. Lett. **63**, 2753 (1989); M. D. Eldridge, P. A. Madden and D. Frenkel, Nature **365**, 35 (1993); N. Hunt, R. Jardine R and P. Bartlett, Phys. Rev. E **62**, 900 (2000); A. Yethiraj and A. van Blaaderen, Nature **421**, 513 (2003).

[8] A. Puglisi, P. Visco, E. Trizac, and F. van Wijland, Phys. Rev. E **73**, 021301 (2006).

[9] F. W. Wiegel and J. P. J. Michels, Chem. Phys. Lett. **40**, 23 (1976).

[10] This problem also pertains to the definition of time in Direct Simulation Monte Carlo algorithms, as discussed in [K. Koura, Phys Fluids **29**, 3509 (1986)].

[11] M. H. Ernst, Phys. Reports **78**, 1 (1981).

[12] C. E. Hecht, Statistical Thermodynamics and Kinetic Theory, Dover edition, New York (1998).

[13] D. L. Blair and A. Kudrolli, Phys. Rev. E **67**, 041301 (2003).

[14] Conversely, the minimum of $r(v)$ is the relevant quantity for the long time behavior of Eq. (4); it is reached at $v = 0$ (see the inset of Fig. 1), where $r(0) = \omega/\sqrt{2}$. This explains the $\sqrt{2}$ factors appearing in Eq. (6).

[15] P. Visco, F. van Wijland and E. Trizac, in preparation.

[16] More precisely, it can be shown that at late times, $\tilde{f}(v, \lambda, t)$ takes the asymptotic form $\exp[\mu(\lambda)t]\tilde{f}(v, \lambda)$, so that the time dependence factorizes from the $v$ variable, which explains why $T(\lambda)$ is $t$-independent. The requirement that $\mu(0) = 0$ ensures that $\tilde{f}(v, 0, t)$ is also time independent.