Conditional probabilities, fluctuation theorem

The world surrounding us is mostly out of equihbrium, equilibrium being just an idealization that requires specific conditions to be met in the laboratory. Even today we do not have a general theory about nonequilibrium macroscopic systems as we have for equilibrium ones. Onsager theory is probably the most successful attempt, albeit its domain of validity is restricted to the linear response regime. In small systems the situation seems to be the opposite. Over the past years, a set of theoretical results that go under the name of fluctuation theorems have been unveiled. These theorems make specific predictions about energy processes in small systems that can be scrutinized in the laboratory. 

Here, <... > denotes an average over the equilibrium ensemble of initial conditions. C t) is the conditional probability to find the system in state B at time t provided it was in state A at time 0. According to the fluctuation-dissipation theorem [63], dynamics of equilibrium fluctuations are equivalent to the relaxation from a nonequilibrium state in which only state A is populated. At long timescales, these nonequilibrium dynamics are described by the phenomenology of macroscopic kinetics. Thus, the asymptotic behavior of C(t) is determined by rate constants and kg. At long times, and provided that a single dynamical bottleneck separating A from B causes simple two-state kinetics. 

At nonequilibrium steady state, a net flux in the species occurs if it is possible to adjust the concentrations. Hence, the stationary state violates the detailed balance condition PnJnm = PmJmn where J is the rate of transformation and p is the probability. For such nonequilibrium steady states, a detailed fluctuation theorem is... 

If they are constants with time, it can be shown (e.g. by using Boltzmann s famous H-theorem) that all distributions developing according to the master equation (3.14) or the Fokker-Planck equation (3.32) or (3.34) end up in the stationary distribution W (jc). After this the motion of a sample socio-configuration can only consist of stationary fluctuations around the probability peaks of the stationary distribution. While this kind of motion may describe a static society under stable psychological and situational conditions, it does not comprise the full scope of sociological phenomena. 

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