Fluctuation statistics in networks: A stochastic path integral approach

Abstract

The modified optimal path method and non-adiabatic II-order transitions in noisy perturbed dynamical systems AIP Conf. Proc. 502, 54 (2000) We investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors. The nodes carry generalized charge that may be randomly transferred from one node to another. Our goal is to find the time evolution of the probability distribution of charges in the network. The building blocks of our theoretical approach are (1) known probability distributions for the connector currents, (2) physical constraints such as local charge conservation, and (3) a time scale separation between the slow charge dynamics of the nodes and the fast current fluctuations of the connectors. We integrate out fast current fluctuations and derive a stochastic path integral representation of the evolution operator for the slow charges. The statistics of charge fluctuations may be found from the saddlepoint approximation of the action. Once the probability distributions on the discrete network have been studied, the continuum limit is taken to obtain a statistical field theory. We find a correspondence between the diffusive field theory and a Langevin equation with Gaussian noise sources, leading nevertheless to nontrivial fluctuation statistics. To complete our theory, we demonstrate that the cascade diagrammatics, recently introduced by Nagaev, naturally follows from the stochastic path integral. By generalizing the principle of minimal correlations, we extend the diagrammatics to calculate current correlation functions for an arbitrary network. One primary application of this formalism is that of full counting statistics (FCS), the motivation for why it was developed in the first place. We stress however, that the formalism is suitable for general classical stochastic problems as an alternative approach to the traditional master equation or Doi-Peliti technique. The formalism is illustrated with several examples: Both instantaneous and time averaged charge fluctuation statistics in a mesoscopic chaotic cavity, as well as the FCS and new results for a generalized diffusive wire.

Fluctuation theorem for heat transport probed by a thermal probe electrode

“…16 As is mentioned above, the temperature of the probe fluctuates in time, i.e., the probe affinity A P is time dependent and consequently so is the (instantaneous) probe energy denoted E(t) which is given by…”

“…The situation is similar to what happens in the presence of gauge fields, where different choices of the gauge result in apparently different expressions, which are, in fact, identical. Previous research has exploited the minimal-correlation coordinate 16 (see Appendix C), which simplifies drastically the calculations at the price of expressions which do not explicitly obey the FT and consequently miss symmetries among the transport coefficients.…”

“…44 In the present paper we adopt a simpler approximation which captures the relevant physics. [15][16][17][18][19][20][21] This approach relies on the existence of two distinct time scales. The faster one pertains to the traveling time of each electron through the conductor and the subsequent relaxation in any of the electrodes.…”

“…For example, the rapid flow of electrons in and out of a voltage-probing electrode results in much slower charge fluctuations there, thus allowing for a stochastic path-integration of the CGF of the full setup (e.g., a three-terminal one) over all configurations of the probe charge, to obtain the reduced CGF of the physical setup (e.g., a two-terminal one). [15][16][17][18] A similar treatment has been carried out for the stochastic temperature and chemical potential fluctuations in an overheated metallic island. [19][20][21] However, to the best of our knowledge there are no studies of the fluctuation theorem (FT) [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] in systems coupled to thermal probes.…”

“…In order to map the three-terminal junction of Fig. 1 onto the effective two-terminal one we adopt the stochastic path-integral formalism, [15][16][17][18] originally devised for describing electric conduction through a chaotic cavity. We analyze the FT pertaining to the resulting effective two-terminal setup.…”