Propagation of Chaos and Effective Equations in Kinetic Theory: a Brief Survey

We review some historical highlights leading to the modern perspective on the concept of chaos from the point of view of the kinetic theory. We focus in particular on the role played by the propagation of chaos in the mathematical derivation of effective equations

New treatments of density fluctuations and recurrence times for re-estimating Zermelo's paradox

What is the probability that all the gas in a box accumulates in the same half of this box? Though amusing, this question underlies the fundamental problem of density fluctuations at equilibrium, which has profound implementations in many physical fields. The currently accepted solutions are derived from the studies of Brownian motion by Smoluchowski, but they are not appropriate for the directly colliding particles of gases. Two alternative theories are proposed here using self-regulatory Bernoulli distributions. A discretization of space is first introduced to develop a mechanism of matter congestion holding for high densities. In a second mechanism valid in ordinary conditions, the influence of local pressure on the location of every particle is examined using classical laws of ideal gases. This approach reveals that a negative feedback results from the reciprocal influences between individual particles and the population of particles, which strongly reduces the probability of atypical microstates. Finally, a thermodynamic quantum of time is defined to compare the recurrence times of improbable macrostates predicted through these different approaches.Comment: Le titre a \'et\'e chang\'e Ancien titre: Roles for local crowding and pressure in counteracting density fluctuations at equilibriu

No information or horizon paradoxes for Th. Smiths

'Th'e 'S'tatistical 'm'echanician 'i'n 'th'e 's'treet (our Th. Smiths) must be surprised upon hearing popular versions of some of today's most discussed paradoxes in astronomy and cosmology. In fact, rather standard reminders of the meaning of thermal probabilities in statistical mechanics appear to answer the horizon problem (one of the major motivations for inflation theory) and the information paradox (related to black hole physics), at least as they are usually presented. Still the paradoxes point to interesting gaps in our statistical understanding of (quantum) gravitational effects

Large deviations of the stochastic area for linear diffusions

The area enclosed by the two-dimensional Brownian motion in the plane was studied by L\'evy, who found the characteristic function and probability density of this random variable. For other planar processes, in particular ergodic diffusions described by linear stochastic differential equations (SDEs), only the expected value of the stochastic area is known. Here, we calculate the generating function of the stochastic area for linear SDEs, which can be related to the integral of the angular momentum, and extract from the result the large deviation functions characterising the dominant part of its probability density in the long-time limit, as well as the effective SDE describing how large deviations arise in that limit. In addition, we obtain the asymptotic mean of the stochastic area, which is known to be related to the probability current, and the asymptotic variance, which is important for determining from observed trajectories whether or not a diffusion is reversible. Examples of reversible and irreversible linear SDEs are studied to illustrate our results.Comment: v1: 13 pages, 7 figures; v2: minor errors corrected; v3: minor edits, close to published versio

Normal transport properties for a classical particle coupled to a non-Ohmic bath

We study the Hamiltonian motion of an ensemble of unconfined classical particles driven by an external field F through a translationally-invariant, thermal array of monochromatic Einstein oscillators. The system does not sustain a stationary state, because the oscillators cannot effectively absorb the energy of high speed particles. We nonetheless show that the system has at all positive temperatures a well-defined low-field mobility over macroscopic time scales of order exp(-c/F). The mobility is independent of F at low fields, and related to the zero-field diffusion constant D through the Einstein relation. The system therefore exhibits normal transport even though the bath obviously has a discrete frequency spectrum (it is simply monochromatic) and is therefore highly non-Ohmic. Such features are usually associated with anomalous transport properties

Deterministic model of battery, uphill currents, and nonequilibrium phase transitions

We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce-back mechanism acts when the number of particles flowing in one direction exceeds a given threshold T. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities of the channels and of T, nonequilibrium phase transitions take place in the N→∞ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant