J. Piasecki

Stationary motion of the adiabatic piston

We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures T1 and T2, separated by an adiabatic movable piston whose mass M is much larger than the mass m of the fluid particles. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter e=m/M, and explicitly given up to order e2. In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the fluctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where M=m.

Aggregation dynamics in a self-gravitating one-dimensional gas

Aggregation of mass by perfectly inelastic collisions in a one-dimensional self-gravitating gas is studied. The binary collisions are subject to the laws of mass and momentum conservation. A method to obtain an exact probabilistic description of aggregation is presented. Since the one-dimensional gravitational attraction is confining, all particles will eventually form a single body. The detailed analysis of the probabilityPn(t) of such a complete merging before timet is performed for initial states ofn equidistant identical particles with uncorrelated velocities. It is found that for a macroscopic amount of matter (n→∞), this probability vanishes before a characteristic timet*. In the limit of a continuous initial mass distribution the exact analytic form ofPn(t) is derived. The analysis of collisions leading to the time-variation ofPn(t), reveals that in fact the merging into macroscopic bodies always occurs in the immediate vicinity oft*. Fort>t*, andn large,Pn(t) describes events corresponding to the final aggregation of remaining microscopic fragments.

Multiple time scale derivation of the Fokker-Planck equation for two Brownian spheres suspended in a hard sphere fluid

The Fokker-Planck equation for the distribution function of two massive Brownian spheres, suspended in a fluid of much lighter spheres, is derived from the full hierarchy of exact kinetic equations for the time evolution of the full system consisting of two Brownian and N fluid spheres. The separation of time scales is automatically achieved by a systematic multiple time-scale analysis of the expansion in powers of the square root of the fluid-to-Brownian particle mass ratio. This procedure guarantees uniform convergence of the expansion and requires no extra physical assumptions to justify the separation of time scales. An exact expression is obtained for the mutual friction tensors, which naturally split into a static (Enskog) part and a contribution due to dynamical correlations. The present derivation of the two-particle Fokker-Planck equation also leads to an expression for the fluid-induced, effective depletion force between two Brownian particles.

From the adiabatic piston to macroscopic motion induced by fluctuations

The controversial problem of the evolution of an isolated system with an internal adiabatic wall is investigated with the use of a simple microscopic model and the Boltzmann equation. In the case of two infinite volume one-dimensional ideal fluids separated by a piston whose mass is equal to the mass of the fluid particles we obtain a rigorous explicit stationary non-equilibrium solution of the Boltzmann equation. It is shown that at equal pressures on both sides of the piston, the temperature difference induces a non-zero average velocity, oriented toward the region of higher temperature. It thus turns out that despite the absence of macroscopic forces the asymmetry of fluctuations results in a systematic macroscopic motion. This remarkable effect is analogous to the dynamics of stochastic ratchets, where fluctuations conspire with spatial anisotropy to generate directed motion. However, a different mechanism is involved here. The relevance of the discovered motion to the adiabatic piston problem is discussed.

On the Brownian Motion of a Massive Sphere Suspended in a Hard-Sphere Fluid. I. Multiple-Time-Scale Analysis and Microscopic Expression for the Friction Coefficient

The Fokker-Planck equation governing the evolution of the distribution function of a massive Brownian hard sphere suspended in a fluid of much lighter spheres is derived from the exact hierarchy of kinetic equations for the total system via a multiple-time-scale analysis akin to a uniform expansion in powers of the square root of the mass ratio. The derivation leads to an exact expression for the friction coefficient which naturally splits into an Enskog contribution and a dynamical correction. The latter, which accounts for correlated collisions events, reduces to the integral of a time-displaced correlation function of dynamical variables linked to the collisional transfer of momentum between the infinitively heavy (i.e., immobile) Brownian sphere and the fluid particles.

Kinetic theory of transport in a hard sphere crystal

The revised Enskog kinetic theory (RET) is used to describe transport in a hard sphere crystal. The connection between the RET and the exact density functional theory (DFT) description of the solid state is established. The RET is used to derive the dissipative linear equations of elasticity. The elastic moduli in these equations are identical to those obtained from equilibrium like DFT. The expressions for the solid state transport coefficients (determining sound absorption and heat conduction in the hard sphere crystal) are new. As for the analogous calculation in the liquid state, the transport coefficients are determined by the (solid state) equilibrium two‐particle distribution function at contact.

Long-Time Behavior of the Lorentz Electron Gas in a Constant, Uniform Electric Field

The long-time behavior of the Lorentz electron gas is studied in the presence of a uniform external field. A discussion of the rigorous solution of the one-dimensional Boltzmann equation is followed by the derivation of the asymptotic form of the velocity distribution in an arbitrary number of dimensions. The system is shown to absorb energy from the field without bounds, which excludes the usually assumed steady state with finite thermal energy density.

Microscopic Derivation of Non-Markovian Thermalization of a Brownian Particle

In this paper, the first microscopic approach to Brownian motion is developed in the case where the mass density of the suspending bath is of the same order of magnitude as that of the Brownian (B) particle. Starting from an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematically via multiple-time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e., the relaxation toward the Maxwell distribution in velocity space. In contradistinction to the Fokker-Planck equation, the derived new evolution equation is nonlocal both in time and in velocity space, owing to correlated recollision events between the fluid and particle B. In the long-time limit, it describes a non-Markovian generalized Ornstein-Uhlenbeck process. However, in spite of this complex dynamical behavior, the Stokes-Einstein law relating the friction and diffusion coefficients is shown to remain valid. A microscopic expression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the meanfree path in the gas is small compared to the radius of particle B.

One-dimensional ballistic aggregation: Rigorous long-time estimates

Aggregation of mass by perfectly inelastic collisions in a one-dimensional gas of point particles is studied. The dynamics is governed by laws of mass and momentum conservation. The motion between collisions is free. An exact probabilistic description of the state of the aggregating gas is presented. For an initial configuration of equidistant particles on the line with Maxwellian velocity distribution, the following results are obtained in the long-time limit. The probability for finding empty intervals of length growing faster thant2/3 vanishes. The mass spectrum can range from the initial mass up to mass of ordert2/3. Aggregates with masses growing faster thant2/3 cannot occur. Our estimates are in accordance with numerical simulations predictingt−1 decay for the number density of initial masses and a slowert−2/3 decay for the density of aggregates resulting from a large number of collisions (with masses ∼t2/3). Our proofs rely on a link between the considered aggregation dynamics and Brownian motion in the presence of absorbing barriers.

From the Liouville Equation to the Generalized Boltzmann Equation for Magnetotransport in the 2D Lorentz Model

We consider a system of non-interacting charged particles moving in two dimensions among fixed hard scatterers, and acted upon by a perpendicular magnetic field. Recollisions between charged particles and scatterers are unavoidable in this case. We derive from the Liouville equation for this system a generalized Boltzmann equation with infinitely long memory, but which still is analytically solvable. This kinetic equation has been earlier written down from intuitive arguments.