Power and Heat Fluctuation Theorems for Electric Circuits

Abstract

Using recent fluctuation theorems from nonequilibrium statistical mechanics, we extend the theory for voltage fluctuations in electric circuits to power and heat fluctuations. They could be of particular relevance for the functioning of small circuits. This is done for a parallel resistor and capacitor with a constant current source for which we use the analogy with a Brownian particle dragged through a fluid by a moving harmonic potential, where circuit-specific analogues are needed on top of the Brownian-Nyquist analogy. The results may also hold for other circuits as another example shows. Nanotechnology is quickly getting within reach, but the physics at these scales could be different from that at the macroscopic scale. In particular, large fluctuations will occur, with mostly unknown consequences. In this letter, we will investigate properties of electric circuits concerning the fluctuations of power and heat within the context of the so-called Fluctuation Theorems (FTs). These theorems were originally found in the context of nonequilibrium dynamical systems theory. Surprisingly, these can be applied also to electric circuits, as we will show, and thus give further insight into their behavior.Let us first give a brief introduction to the FTs. First found in dynamical systems[1, 2] and later extended to stochastic systems[3], these conventional FTs give a relation between the probabilities to observe a positive value of the (time averaged) "entropy production rate" and a negative one. This relation is of the form P (σ)/P (−σ) = exp [στ ], where σ and −σ are equal but opposite values for the entropy production rate, P (σ) and P (−σ) give their probabilities and τ is the length of the interval over which σ is measured. In these systems, the above mentioned FT is derived for a mathematical quantity σ, which has a form similar to that of the entropy production rate in Irreversible Thermodynamics.Apart from an early experiment in a turbulent flow [4], for quite some time, the investigations of the FTs were restricted to theoretical approaches and simulations. In 2002, Wang et al. performed an experiment on a micron-sized Brownian particle dragged through water by a moving optical tweezer. In this experiment, a Transient Fluctuation Theorem (TFT) was demonstrated for fluctuations of the total external work done on the system in the transient state of the system, i.e., considering a time interval of duration τ which starts immediately after the tweezer has been set in motion [5]. In contrast, a Stationary State Fluctuation Theorem (SSFT), which was not measured, would concern fluctuations in the stationary state, i.e., in intervals of duration τ starting at a time long after the tweezer has been set in motion. While the work fluctuations satisfy the conventional TFT and SSFT [6,7], the heat fluctuations satisfy different, extended FTs due to the interplay of the stochastic motion of the fluid with the deterministic harmonic potential induced by the optical tweezer [8,9]. Given the possible problems with identifyin...

Chaotic Hypothesis, Fluctuation Theorem and Singularities

“…it is of the form ∼ e −κF , for some constant κ > 0. This is a valuable and interesting remark brought up for the first time, and correctly interpreted, already in [4] and in the following papers [24,25]. The analysis of [4,24,25] applies to cases where the unbounded fluctuations are driven by an external white noise.…”

“…The prediction that (at least near equilibrium) the rate function of a should satisfy FR only up to a = σ + and that should become linear for a ≥ a + at the moment has been experimentally confirmed only in Gaussian cases [4,24,25]. It would be very interesting to investigate in detail the structure of ζ(a) even in non Gaussian cases.…”

“…The analysis in Sect. 2 above applies as well to understand how to apply the FR to systems with Gaussian (or unbounded) noise and the compatibility between the general theory of [31,32,37] with the works [24] and [25,38].…”

“…The analysis of [4,24,25] applies to cases where the unbounded fluctuations are driven by an external white noise. In the following we extend the theoretical analysis in [24,25] to cases in which the unbounded fluctuations do not arise from a Gaussian noise but from a deterministic evolution like the ones in [3,15]: this is a simple extension of the main idea and method of [24] and provides an alternative interpretation to the analysis in [3,15].…”

Exponential peak and scaling of work fluctuations in modulated systems

“…A major "test bed" for fluctuation theorems is provided by dynamical systems with a few degrees of freedom coupled to a thermal bath, a Brownian particle being an example. Much of the corresponding theoretical and experimental work refers to (i) modulated linear systems, where fluctuations have been studied both in transient and stationary regimes [6,7,8,9,10,11,12], and (ii) nonlinear systems, initially at thermal equilibrium, driven to a different, generally nonequilibrium state [13,14,15,16,17].…”