An Investigation Into the Significance of Dissipation in Statistical Mechanics

Abstract

The dissipation function is a key quantity in nonequilibrium statistical mechanics. It was originally derived for use in the Evans-Searles Fluctuation Theorem, which quantitatively describes thermal fluctuations in nonequilibrium systems. It is now the subject of a number of other exact results, including the Dissipation Theorem, describing the evolution of a system in time, and the Relaxation Theorem, proving the ubiquitous phenomena of relaxation to equilibrium. The aim of this work is to study the significance of the dissipation function, and examine a number of exact results for which it is the argument. First, we investigate a simple system relaxing towards equilibrium, and use this as a medium to investigate the role of the dissipation function in relaxation. The initial system has a non-uniform density distribution. We demonstrate some of the existing significant exact results in nonequilibrium statistical mechanics. By modifying the initial conditions of our system we are able to observe both monotonic and non-monotonic relaxation towards equilibrium. A direct result of the Evans-Searles Fluctuation Theorem is the Nonequilibrium Partition Identity (NPI), an ensemble average involving the dissipation function. While the derivation is straightforward, calculation of this quantity is anything but. The statistics of the average are difficult to work with because its value is extremely dependent on rare events. It is often observed to converge with high accuracy to a value less than expected. We investigate the mechanism for this asymmetric bias and provide alternatives to calculating the full ensemble average that display better statistics. While the NPI is derived exactly for transient systems it is expected that it will hold in steady state systems as well. We show that this is not true, regardless of the statistics of the calculation. A new exact result involving the dissipation function, the Instantaneous Fluctuation Theorem, is derived and demonstrated computationally. This new theorem has the same form as previous fluctuation theorems, but provides information about the instantaneous value of phase functions, rather than path integrals. We extend this work by deriving an approximate form of the theorem for steady state systems, and examine the validity of the assumptions used.