On the derivation of Fourier's law in stochastic energy exchange systems

We present a detailed derivation of Fourier's law in a class of stochastic energy exchange systems that naturally characterize two-dimensional mechanical systems of locally confined particles in interaction. The stochastic systems consist of an array of energy variables which can be partially exchanged among nearest neighbours at variable rates. We provide two independent derivations of the thermal conductivity and prove this quantity is identical to the frequency of energy exchanges. The first derivation relies on the diffusion of the Helfand moment, which is determined solely by static averages. The second approach relies on a gradient expansion of the probability measure around a non-equilibrium stationary state. The linear part of the heat current is determined by local thermal equilibrium distributions which solve a Boltzmann-like equation. A numerical scheme is presented with computations of the conductivity along our two methods. The results are in excellent agreement with our theory.Comment: 19 pages, 5 figures, to appear in Journal of Statistical Mechanics (JSTAT

On the Notion of Pressure in a Canonical Ensemble

Recently a controversy has arisen between J. De Boer and H. S. Green concerning the notion of pressure in a canonical ensemble. According to Green, only classically is the pressure as derived from the partition function equal to that obtained from the virial theorem, while at low temperatures, at which quantum effects become important, there will be considerable deviations between the two. De Boer attempts to prove that the two pressures are actually identical. We have come to the same conclusion, and shall show this in several ways; first, by considering a simple example (Section 2), and then in general using the energy representation (Section 3). We believe that the discrepancy between the two pressures which Green has found is in fact due to improper handling of the effect of the wall of the vessel in which the particles are contained. Finally, we are of the opinion that Green's criticism of De Boer's calculation is not justified, and in the last section arguments are given to show that the traces of all commutators of interest in quantum statistical mechanics are zero.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69356/2/JCPSA6-18-8-1066-1.pd

On the Theory of the Virial Development of the Equation of State of Monoatomic Gases

The problem of the condensation of a gas is intimately related to the asymptotic behavior of the virial coefficients, Bm, as m→∞. The problem of the evaluation of the virial coefficients may be divided into two distinctly different ones. The first of these, which is purely combinatorial in nature and is independent of the intermolecular force law, is that of determining the number of a certain type of connected graphs of l points and k lines which are called ``stars.'' This problem is solved by means of generating functions, with the result that the total number of such stars is asymptotically equal to(12l(l−1)k),for almost all k. Arguments are also presented which indicate that the total number of topologically different stars is1l!(12l(l−1)k).With these results the combinatorial problem is essentially solved.The second problem is that of evaluating certain integrals of functions which depend on the intermolecular potential. This problem is not so near to a solution. For a purely repulsive force, asymptotic expressions are obtained for k=l, and k=l+1. The partial contributions to the virial coefficient in these two cases are:(−1)l⋅53(52π)12(2b)l−1(l−1)l5∕2,and(−1)l2⋅5324π3(2b)l−1,respectively. Results for some simple one‐dimensional rigid lines are also given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69794/2/JCPSA6-21-11-2056-1.pd

The kinetic method in statistical mechanics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32617/1/0000760.pd

Successive approximation methods in classical statistical mechanics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32400/1/0000475.pd

Perturbation theory for a stochastic process with Ornstein-Uhlenbeck noise

The Ornstein-Uhlenbeck process may be used to generate a noise signal with a finite correlation time. If a one-dimensional stochastic process is driven by such a noise source, it may be analysed by solving a Fokker-Planck equation in two dimensions. In the case of motion in the vicinity of an attractive fixed point, it is shown how the solution of this equation can be developed as a power series. The coefficients are determined exactly by using algebraic properties of a system of annihilation and creation operators.Comment: 7 pages, 0 figure

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

Fluid/solid transition in a hard-core system

We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition

Quantum many-body theory of qubit decoherence in a finite-size spin bath

Decoherence of a center spin or qubit in a spin bath is essentially determined by the many-body bath evolution. We develop a cluster-correlation expansion (CCE) theory for the spin bath dynamics relevant to the qubit decoherence problem. A cluster correlation term is recursively defined as the evolution of a group of bath spins divided by the cluster correlations of all the subgroups. The so-defined correlation accounts for the authentic collective excitations within a given group. The bath propagator is the product of all possible cluster correlation terms. For a finite-time evolution as in the qubit decoherence problem, a convergent result can be obtained by truncating the expansion up to a certain cluster size. The two-spin cluster truncation of the CCE corresponds to the pair-correlation approximation [PRB 74,195301(2006)]. In terms of the standard linked cluster expansion, a cluster correlation term is the infinite summation of all the connected diagrams with all and only the spins in the group flip-flopped, and thus the expansion is exact whenever converges. When the individual contribution of each higher-order correlation term to the decoherence is small, as the usual case for relatively large baths where the decoherence could complete well within the bath spin flip-flop time, the CCE coincides with the cluster expansion [PRB 74,035322(2006)]. For small baths, however, the qubit decoherence may not complete within the bath spin flip-flop timescale and thus individual higher-order cluster correlations could grow significant. In such cases, only the CCE converges to the exact coherent dynamics of multi-spin clusters. We check the accuracy of the CCE in an exactly solvable spin-chain model.Comment: 13 pages, 13 figure

Comparison of work fluctuation relations

We compare two predictions regarding the microscopic fluctuations of a system that is driven away from equilibrium: one due to Crooks [J. Stat. Phys. 90, 1481 (1998)] which has gained recent attention in the context of nonequilibrium work and fluctuation theorems, and an earlier, analogous result obtained by Bochkov and Kuzovlev [Zh. Eksp. Teor. Fiz. 72(1), 238247 (1977)]. Both results quantify irreversible behavior by comparing probabilities of observing particular microscopic trajectories during thermodynamic processes related by time-reversal, and both are expressed in terms of the work performed when driving the system away from equilibrium. By deriving these two predictions within a single, Hamiltonian framework, we clarify the precise relationship between them, and discuss how the different definitions of work used by the two sets of authors gives rise to different physical interpretations. We then obtain a extended fluctuation relation that contains both the Crooks and the Bochkov-Kuzovlev results as special cases.Comment: 14 pages with 1 figure, accepted for publication in the Journal of Statistical Mechanic