Hashitsume

Kubo R, Toda M and Hashitsume N 1985 Statistical Physics //(Berlin Springer) ch 4... 

R. Kubo, M. Toda, N. Hashitsume. Statistical Physics II. Berlin Springer, 1985. 

On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

In the work of Brown [5] and Kubo and Hashitsume [45] the starting equation is the Gilbert equation (3.43), in which the effective field is increased by a fluctuating field yielding the stochastic Gilbert equation. This equation can, as in the deterministic case, be cast into the Landau-Lifshitz form as... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

Plugging this result into the differential equation for the relevant part, Eq. (50), the Hashitsume-Shibata-Takahashi identity can be obtained... 

Kubo, R., Toda, M., Hashitsume, N. Statistical physics II non equilibrium statistical mechanics. Ed. Springer-Verlag (1985). 

Kubo R, Toda M, Hashitsume N. Statistical Physics II Nonequilibrium Statistical Mechanics. 2nd ed. Berlin Springer, 1991. 

D. Chandler, Introduction to Modem Statistical Mechanics (Oxford University Press, Oxford, 1986). R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics 11, Springer Series in Solid State Sciences,... 

Kubo, R., 1968, Thermodynamics An Jidvaiiced course with problems and. solutions (in cooperation with 11. Ic.himura, T. Usui, N. Hashitsume) (North-Holland, Amsterdam). 

This transition probability was first obtained by Onsager and the probability for the path is obtained from Eq. 50. Hashitsume and Onsager and Machlup used extensively this probability of path in the theory of thermal fluctuations. 

Kubo, R., Ichimura, H., Usui, T., and Hashitsume, N. (1965) Statistical Mechanics An Advanced Course with Problems and Solutions, North-Holland, Amsterdam. 

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