Non-equilibrium Thermodynamics and Fluctuations

In the last ten years, a number of ``Conventional Fluctuation Theorems'' have been derived for systems with deterministic or stochastic dynamics, in a transient or in a non-equilibrium stationary state. These theorems gave explicit expressions for the ratio of the probability to find the system with a certain value of entropy (or heat) production to that of finding the opposite value. A similar theorem for the fluctuations of the work done on a system has recently been demonstrated experimentally for a simple system in a transient state, consisting of a Brownian particle in water, confined by a moving harmonic potential. In this paper we show that because of the interaction between the stochastic motion of the particle in water and its deterministic motion in the potential, very different new heat theorems are found than in the conventional case. One of the consequences of these new heat Fluctuation Theorems is that the ratio of the probability for the Brownian particle to absorb heat from rather than supply heat to the water is much larger than in the Conventional Fluctuation Theorem. This could be of relevance for micro/nano-technology.Comment: 10 pages, 6 figures. Some corrections in the text were made. Submitted to Physica

Long time diffusion in suspensions of interacting charged colloids

A new expression is given for the long time diffusion coefficient DL(k) of charged interacting colloidal spheres in suspension, as a function of the wavenumber k, near k = km, where the static structure factor has a maximum. The expression is based on a physical analogy between a mode description of the behaviour of atomic fluids (as observed in neutron scattering) and of colloids (as observed in light scattering). Use of this expresssion in conjunction with a hard-sphere model yields good agreement with extant data on colloids

Short-time fluctuations of displacements and work

A recent theorem giving the initial behavior of very short-time fluctuations of particle displacements in classical many-body systems is discussed. It has applications to equilibrium and non-equilibrium systems, one of which is a series expansion of the distribution of work fluctuations around a Gaussian function. To determine the time-scale at which this series expansion is valid, we present preliminary numerical results for a Lennard-Jones fluid. These results suggest that the series expansion converges up to time scales on the order of a picosecond, below which a simple Gaussian function for the distribution of the displacements can be used.Comment: 10 pages, 3 figure

Superstatistics

We consider nonequilibrium systems with complex dynamics in stationary states with large fluctuations of intensive quantities (e.g. the temperature, chemical potential, or energy dissipation) on long time scales. Depending on the statistical properties of the fluctuations, we obtain different effective statistical mechanics descriptions. Tsallis statistics is one, but other classes of generalized statistics are obtained as well. We show that for small variance of the fluctuations all these different statistics behave in a universal way.Comment: 12 pages /a few more references and comments added in revised versio

Theorem on the Distribution of Short Time Single Particle Displacements

The distribution of the initial very short-time displacements of a single particle is considered for a class of classical systems with Gaussian initial velocity distributions and arbitrary initial particle positions. A very brief sketch is given of a rather intricate and lengthy proof that for this class of systems the nth order cumulants behave as t^{2n} for all n>2, rather than as t^{n}. We also briefly discuss some physical consequences for liquids.Comment: Short 8 page pedagogical review of cond-mat/0505734 for Proc. of "News, Expectations and Trends in Statistical Physics", Crete 200

On first-order phase transition in microcanonical and canonical non-extensive systems

Two examples of Microcanonical Potts models, 2-dimensional nearest neighbor and mean field, are considered via exact enumeration of states and analytical asymptotic methods. In the interval of energies corresponding to a first order phase transition, both of these models exhibit a convex dip in the entropy vs energy plot and a region with negative specific heat within the dip. It is observed that in the nearest neighbor model the dip flattens and disappears as the lattice size grows, while in the mean field model the dip persists even in the limit of an infinite system. If formal transitions from microcanonical to canonical ensembles and back are performed for an infinite but non-extensive system, the convex dip in the microcanonical entropy plot disappears.Comment: 10 pages, 8 figure

Superstatistical generalization of the work fluctuation theorem

We derive a generalized version of the work fluctuation theorem for nonequilibrium systems with spatio-temporal temperature fluctuations. For chi-square distributed inverse temperature we obtain a generalized fluctuation theorem based on q-exponentials, whereas for other temperature distributions more complicated formulae arise. Since q-exponentials have a power law decay, the decay rate in this generalized fluctuation theorem is much slower than the conventional exponential decay. This implies that work fluctuations can be of relevance for the design of micro and nano structures, since the work done on the system is relatively much larger than in the conventional fluctuation theorem.Comment: 13 pages. Contribution to the Proceedings of `Trends and Perspectives in Extensive and Nonextensive Statistical Mechanics', in honour of Constantino Tsallis' 60th birthday (to appear in Physica A

Stationary and Transient Work-Fluctuation Theorems for a Dragged Brownian Particle

Recently Wang et al. carried out a laboratory experiment, where a Brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center. This experiment confirmed a theoretically predicted work related integrated (I) Transient Fluctuation Theorem (ITFT), which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state. The corresponding integrated stationary state fluctuation theorem (ISSFT) was not observed. Using an overdamped Langevin equation and an arbitrary motion for the center of the harmonic force, all quantities of interest for these theorems and the corresponding non-integrated ones (TFT and SSFT, resp.) are theoretically explicitly obtained in this paper. While the (I)TFT is satisfied for all times, the (I)SSFT only holds asymptotically in time. Suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the ISSFT could be observed, are given. In addition, a non-trivial long-time relation between the ITFT and the ISSFT was discovered, which could be observed experimentally, especially in the case of a resonant circular motion of the center of the harmonic force.Comment: 20 pages, 3 figure

Hydrodynamics of probabilistic ballistic annihilation

We consider a dilute gas of hard spheres in dimension $d \geq 2$ that upon collision either annihilate with probability $p$ or undergo an elastic scattering with probability $1-p$. For such a system neither mass, momentum, nor kinetic energy are conserved quantities. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine the transport coefficients up to Navier-Stokes order, and give the closed set of equations for the hydrodynamic fields chosen for the above coarse grained description (density, momentum and kinetic temperature). Linear stability analysis is performed, and the conditions of stability for the local fields are discussed.Comment: 19 pages, 3 eps figures include

Fluctuation formula for nonreversible dynamics in the thermostated Lorentz gas

We investigate numerically the validity of the Gallavotti-Cohen fluctuation formula in the two and three dimensional periodic Lorentz gas subjected to constant electric and magnetic fields and thermostated by the Gaussian isokinetic thermostat. The magnetic field breaks the time reversal symmetry, and by choosing its orientation with respect to the lattice one can have either a generalized reversing symmetry or no reversibility at all. Our results indicate that the scaling property described by the fluctuation formula may be approximately valid for large fluctuations even in the absence of reversibility.Comment: 6 pages, 6 figure