Fluctuating currents in stochastic thermodynamics. I. Gauge invariance of asymptotic statistics

Artur Wachtel, Jürgen Vollmer, and Bernhard Altaner

Phys. Rev. E 92, 042132 – Published 14 October 2015

Abstract

Stochastic thermodynamics uses Markovian jump processes to model random transitions between observable mesoscopic states. Physical currents are obtained from antisymmetric jump observables defined on the edges of the graph representing the network of states. The asymptotic statistics of such currents are characterized by scaled cumulants. In the present work, we use the algebraic and topological structure of Markovian models to prove a gauge invariance of the scaled cumulant-generating function. Exploiting this invariance yields an efficient algorithm for practical calculations of asymptotic averages and correlation integrals. We discuss how our approach generalizes the Schnakenberg decomposition of the average entropy-production rate, and how it unifies previous work. The application of our results to concrete models is presented in an accompanying publication.

Received 4 May 2015

DOI:https://doi.org/10.1103/PhysRevE.92.042132

Authors & Affiliations

Artur Wachtel^*, Jürgen Vollmer, and Bernhard Altaner

Department of Dynamics of Complex Fluids (DCF), Max Planck Institute for Dynamics and Self-Organization (MPI DS), Am Fassberg 17, 37077 Göttingen, Germany and Institute for Nonlinear Dynamics, Faculty of Physics, Georg-August University Göttingen, 37077 Göttingen, Germany

^*Present address: Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, Luxembourg.

Fluctuating currents in stochastic thermodynamics. II. Energy conversion and nonequilibrium response in kinesin models

Bernhard Altaner, Artur Wachtel, and Jürgen Vollmer

Phys. Rev. E 92, 042133 (2015)

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Vol. 92, Iss. 4 — October 2015

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Images

Figure 1
An example graph representing the network of states of a dynamically reversible Markov process, as used in models of stochastic thermodynamics. It is represented either as (a) a graph $(V, E_{d})$ with two directed edges connecting each pair of states, or (b) by choosing a reference orientation as a simple graph $(V, E_{o})$ . We will use the latter throughout this text.
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Figure 2
The action of the boundary and the co-boundary exemplified using the graph of Fig. 1. The reference orientation is omitted here. (a) The boundary operator $\partial$ maps an edge (left) to a linear combination of the two vertices that are connected by the edge (right). (b) A linear combination of edges that does not have any sinks or sources has boundary zero—the defining property of a cycle; see the main text. (c) The co-boundary operator $\partial^{*}$ maps a single vertex (left) to the sum of edges pointing away from that vertex (right). The coefficients for the linear combinations of edges and vertices are encoded as indicated at the bottom right. Note that a negative coefficient associated to an edge is equivalent to a positive coefficient with reversed orientation.
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Figure 3
Three different spanning trees for the oriented graph given in Fig. 1. The edges $T$ of the different trees are marked green (gray), while the chords $H$ are depicted in dark blue (dark gray). Up to symmetries, every other spanning tree of the graph looks like one of the depicted ones.
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Figure 4
Fundamental cycles of the spanning trees in Fig. 3: every chord $η \in H$ (dashed here, dark blue in Fig. 3) generates a fundamental cycle, that is also marked in dark blue (dark gray) here. The light gray edges are not part of the fundamental cycles. Note that the spanning tree in Fig. 3 generates the exact same fundamental cycles as the one in Fig. 3, but with different chords. In contrast, the spanning tree in Fig. 3 shares only one fundamental cycle with the spanning trees (b) and (c).
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Figure 5
Geometrical interpretation of the chord representative: Projecting $φ$ in parallel to $Y$ onto the chord space $R^{H}$ gives $φ_{H}$ .
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Figure 6
Fundamental co-cycles of the spanning tree in Fig. 3: removing an edge $τ \in T$ (dashed) from the spanning tree, decomposes the graph into two disconnected components marked in dark and light gray, respectively. The green arrows form the corresponding fundamental co-cycle: they are the set of edges pointing from one component to the other.
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