# Stochastic path integral

Stochastic path integral, introduced in some of the papers below, allows calculation of particle transport rates between nodes in a large network (biochemical in our case, and electron transport for when it was originally developed). The jist of the method is to assume Poisson transition among all nodes on short time scales, partition time into small intervals, over which many transitions may happen, but the transition still remain Poissonian, and integrate over trajectories of concentrations as a function of time.

## References

- AN Jordan, EV Sukhorukov, and S Pilgram. Fluctuation statistics in networks: A stochastic path integral approach.
*J Math Phys***45**:4386, 2004. PDF.- Abstract
- 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.

- S Pilgram, AN Jordan, EV Sukhorukov, and M Buttiker. Stochastic Path Integral Formulation of Full Counting Statistics.
*PRL***90**:206801, 2003. PDF.- Abstract
- We derive a stochastic path integral representation of counting statistics in semiclassical systems. The formalism is introduced on the simple case of a single chaotic cavity with two quantum point contacts, and then further generalized to find the propagator for charge distributions with an arbitrary number of counting fields and generalized charges. The counting statistics is given by the saddle-point approximation to the path integral, and fluctuations around the saddle point are suppressed in the semiclassical approximation. We use this approach to derive the current cumulants of a chaotic cavity in the hotelectron regime.

- V Elgart and A Kamenev. Rare event statistics in reaction-diffusion systems.
*PRE***70**:041106, 2004. PDF.- Abstract
- We present an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction-diffusion systems. This method is based on a semiclassical treatment of an underlying “quantum” Hamiltonian, encoding the system’s evolution. To this end, we formulate the corresponding canonical dynamical system and investigate its phase portrait. This method is presented for a number of pedagogical examples.

- V Elgart and A Kamenev. Classification of phase transitions in reaction-diffusion models.
*PRE***74**:041101, 2006. PDF.- Abstract
- Equilibrium phase transitions are associated with rearrangements of minima of a Lagrangian potential. Treatment of nonequilibrium systems requires doubling of degrees of freedom, which may be often interpreted as a transition from the “coordinate”- to the “phase”-space representation. As a result, one has to deal with the Hamiltonian formulation of the field theory instead of the Lagrangian one. We suggest a classification scheme of phase transitions in reaction-diffusion models based on the topology of the phase portraits of corresponding Hamiltonians. In models with an absorbing state such a topology is fully determined by intersecting curves of zero “energy.” We identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations.