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

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

Successive approximation methods in classical statistical mechanics

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

Slow imbalance relaxation and thermoelectric transport in graphene

We compute the electronic component of the thermal conductivity (TC) and the thermoelectric power (TEP) of monolayer graphene, within the hydrodynamic regime, taking into account the slow rate of carrier population imbalance relaxation. Interband electron-hole generation and recombination processes are inefficient due to the non-decaying nature of the relativistic energy spectrum. As a result, a population imbalance of the conduction and valence bands is generically induced upon the application of a thermal gradient. We show that the thermoelectric response of a graphene monolayer depends upon the ratio of the sample length to an intrinsic length scale l_Q, set by the imbalance relaxation rate. At the same time, we incorporate the crucial influence of the metallic contacts required for the thermopower measurement (under open circuit boundary conditions), since carrier exchange with the contacts also relaxes the imbalance. These effects are especially pronounced for clean graphene, where the thermoelectric transport is limited exclusively by intercarrier collisions. For specimens shorter than l_Q, the population imbalance extends throughout the sample; the TC and TEP asymptote toward their zero imbalance relaxation limits. In the opposite limit of a graphene slab longer than l_Q, at non-zero doping the TC and TEP approach intrinsic values characteristic of the infinite imbalance relaxation limit. Samples of intermediate (long) length in the doped (undoped) case are predicted to exhibit an inhomogeneous temperature profile, whilst the TC and TEP grow linearly with the system size. In all cases except for the shortest devices, we develop a picture of bulk electron and hole number currents that flow between thermally conductive leads, where steady-state recombination and generation processes relax the accumulating imbalance.Comment: 14 pages, 4 figure

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

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

The linear Fokker-Planck equation for the Ornstein-Uhlenbeck process as an (almost) nonlinear kinetic equation for an isolated N-particle system

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the infinite particle limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation.Comment: Minor revisions and corrections (including the title