A Gaussian steady states distributions

Thermal response of nonequilibrium RC-circuits

Abstract

We analyze experimental data obtained from an electrical circuit having components at different temperatures, showing how to predict its response to temperature variations. This illustrates in detail how to utilize a recent linear response theory for nonequilibrium overdamped stochastic systems. To validate these results, we introduce a reweighting procedure that mimics the actual realization of the perturbation and allows extracting the susceptibility of the system from steady state data. This procedure is closely related to other fluctuation-response relations based on the knowledge of the steady state probability distribution. As an example, we show that the nonequilibrium heat capacity in general does not correspond to the correlation between the energy of the system and the heat flowing into it. Rather, also non-dissipative aspects are relevant in the nonequilbrium fluctuation response relations.

pacs:
05.40.-a, 05.70.Ln

Introduction

Understanding how a system responds to variations of its parameters is one of the basic features of science. It is well known that systems in thermodynamic equilibrium when slightly perturbed find their way back to a new steady regime by dissipation. The spontaneous correlations in the unperturbed system between this transient entropic change and an observable anticipates us how that observable would react to the actual perturbation. This is at the basis of the fluctuation-dissipation theorem and of related response relations, which hold in great generality in equilibrium Kubo et al. (1992); Marini Bettolo Marconi et al. (2008).

Out of equilibrium, in contrast, there are multiple linear response theories Marini Bettolo Marconi et al. (2008); Baiesi and Maes (2013); Basu and Maes (2015), some based on the manipulation of the density of states Agarwal (1972); Falcioni et al. (1990); Speck and Seifert (2009); Seifert and Speck (2010); Prost et al. (2009), some on dynamical systems techniques for evolving observables Ruelle (2009); Colangeli et al. (2012); Colangeli and Lucarini (2014), and some on a path-weight approach for stochastic systems Cugliandolo et al. (1994); Chetrite et al. (2008); Verley et al. (2011); Lippiello et al. (2005); Baiesi et al. (2009). The latter has revealed that entropy production is not sufficient for understanding the linear response of nonequilibrium systems. There are rather non-dissipative aspects of the system vs perturbation relation that are equally relevant.

Within the linear response theory one finds recent approaches focusing on temperature perturbations Chetrite (2009); Boksenbojm et al. (2011); Baiesi et al. (2014); Brandner et al. (2016); Ford et al. (2015); Proesmans et al. (2016); Yolcu and Baiesi (2016); Falasco and Baiesi (2016a, b), which lead for instance to a formulation of nonequilibrium heat capacity Boksenbojm et al. (2011), a notion that should be useful for constructing a steady state thermodynamics Sekimoto (2007); Sagawa and Hayakawa (2011); Bertini et al. (2013); Maes and Netočný (2014); Komatsu et al. (2015). The question is how a system far from equilibrium reacts to a change of one or many of its bath temperatures. For example, one could be interested in the response to temperature variations of a glassy system undergoing a relaxation process Mauro et al. (2010); Gomez-Solano et al. (2012). Alternatively, a nonequilibrium steady state may be imposed by putting the system in contact with two reservoirs at different temperatures Ciliberto et al. (2013); Bérut et al. (2014). It is the case of an experiment recently realized with a simple desktop electric circuit in which one resistor was kept at room temperature while the other was maintained at a lower temperature Ciliberto et al. (2013).

In this paper, we analyze the experiments of the thermally unbalanced electric circuit Ciliberto et al. (2013) and we discuss the good performances of a fluctuation-response relation Falasco and Baiesi (2016a, b) in computing the susceptibility of the system to a change of the colder, manipulable temperature. The primary goal of this work is to show how to apply this approach in practice. This is a stand-alone procedure for predicting the thermal linear response of the system. Just to validate its results, we compare them with an alternative estimate of the susceptibility, which is introduced here to exploit the knowledge of the steady state data (which is accessible for the simple system analyzed), used by us in a reweighted form to replace the actual application of the perturbation. This useful procedure, as a byproduct, constitutes a new result of this work. We also show the connection of this reweighting procedure with another fluctuation-response relation based on the steady state distribution, put forward by Seifert and Speck Seifert and Speck (2010).

In the following section we describe the experimental setup, then we recall the structure of the fluctuation-response relation and we specialize it to our system. In Sec. III we introduce the reweighting procedure, and in Sec. IV we show how to compute a nonequilibrium version of the heat capacity. The conclusions are followed by an appendix in which we recall in detail the steps to compute the Gaussian steady state distribution of linear stable systems and we specify its form for the electrical circuit.

Figure 1: (Color online) (a) Diagram of the circuit. The resistor is kept at temperature while is always at room temperature . They are coupled via the capacitor . The capacitors and account for the capacitance of the wiring, etc. The voltage generators and represent thermal fluctuations of the voltage that the resistors undergo. (b) Equivalent mechanical system with two Brownian particles moving in fluids at different temperatures and , but trapped and coupled by harmonic springs.

I Experimental set-up

Our experimental set-up is sketched in Fig.1(a). It is constituted by two resistors and , which are kept at different temperature, and , respectively. These temperatures are controlled by thermal baths and is kept fixed at whereas can be set at a value between and using the stratified vapor above a liquid nitrogen bath. The coupling capacitor controls the electrical power exchanged between the resistors and as a consequence the energy exchanged between the two baths. No other coupling exists between the two resistors, which are inside two separated screened boxes. The quantities and are the capacitances of the circuits and the cables. Two extremely low noise amplifiers and Cannatà et al. (2009) measure the voltage and across the resistors and , respectively. All the relevant quantities considered in this paper can be derived by the measurements of and , as discussed below. In particular, the relationships between the measured voltages and the charges are

(1)
(2)

Assuming an initially neutral circuit, we denote by the charge that has flown through the resistor into the node at potential , and by the charge that has flown through out of the node at . By analyzing the circuit one finds that the equations of motion for these charges are

(3a)
(3b)

where

(4)

and is a white noise satisfying . Indeed, in Fig.1(a) the two resistances have been drawn with their associated thermal noise generators and , whose power spectral densities are given by the Nyquist formula , with .

More details on the experimental set-up can be found in Ref. Ciliberto et al. (2013). For the data used for the analysis discussed in the following section, the values of the components are: pF, pF, pF and M. The longest characteristic time of the system is , which for the mentioned values of the parameters is ms.

Ii Thermal response

The system has degrees of freedom. Eqs. (3) can be mapped onto the mechanical system in Fig. 1(b) involving two Brownian particles coupled by harmonic springs,

(5a)
(5b)

Here are the two positions with when the springs are at rest, the temperatures, and the (harmonic) forces are derived from the potential

(6)

The detailed mapping between the electrical and mechanical models is summarized in Table 1; for instance, the admittance is mapped to the mobility . Again, each Gaussian white noise is uncorrelated from the other,

(7)

This recasting in the form (5) allows us to use some recently introduced thermal response formulas Falasco and Baiesi (2016a, b). They predict the linear response of an overdamped stochastic system with additive noise, in general nonequilibrium conditions, when the perturbation is a change of one or more temperatures. In accordance with the presentation of that approach, we choose natural units () in the following, taking temperatures to have dimensions of energy.

electrical mechanical
(mobility)
(spring constant)
Table 1: Mapping between electric quantities and mechanical ones. Note the inversion of indices for and .

The thermal susceptibility of a state observable is defined as the response to a step variation of the set of temperatures, parametrized by a function for times and constant for . In particular, with indicators () that specify which temperatures receive the perturbation, here we write

(8)

The susceptibility as a function of time is then

(9)

In the averages, the first subscript represents the initial () temperatures, while the second subscript represents the temperatures under which the observed dynamics () takes place.

A recent fluctuation-response relation Falasco and Baiesi (2016a, b) expresses the susceptibility (9) of the state observable as a sum,

(10)

where the terms are

(11a)
(11b)
(11c)
(11d)

with the shorthand , and denoting unperturbed averages which have an understood dependence on the distribution at the time when the perturbation is turned on. (Let us stress that the labels of these and terms have nothing to do with the index of the resistors, particles, etc.) Integrals are in the Stratonovich sense, hence in their discretized version one performs midpoint averages, such as . (However, temperatures and mobilities do not depend on the coordinates and the interpretation of the stochastic equation is free.)

The term is a standard correlation between observable and entropy production, but it contains a prefactor not present in the equilibrium version (Kubo formula). The term instead correlates the observable with another form of entropy production and clearly it is relevant only if for some . The terms and , previously called the frenetic terms Baiesi and Maes (2013); Baiesi et al. (2014); Yolcu and Baiesi (2016); Falasco and Baiesi (2016a, b), instead correlate the observable with time-symmetric aspects of the dynamics. These are necessarily non-dissipative in nature. In all cases it is understood that we are dealing with quantities in excess due to the perturbation. The generalized generator

(12)

was introduced to describe the evolution of the degrees of freedom in terms of the -th thermal time as dictated by the -th temperature (see Falasco and Baiesi (2016b) for more details). It differs from the backward generator of the dynamics (5),

(13)

whose action on a state function inside an average is expressed as . The definition of a thermal time permits to recast (5) as isothermal dynamics. For example, if and hence is taken as a reference, then the thermal time yields for

(14)

where the different intensity of the noise (it has now in the prefactor) is associated with a rescaling of the mechanical force .

In our analysis we work with experimental trajectory data collected in steady states, where for . Hence we rather use the alternative form

(15)

because it is numerically more stable than  Falasco and Baiesi (2016b). Since in the given experimental setup it is natural to manipulate (while the room temperature, , remains unperturbed), we show examples with and . This leads to the susceptibility being composed of the specific terms

(16a)
(16b)
(16c)
(16d)

with . Note that we have dropped the superscript “s” from .

The susceptibility is found as the sum of these correlations with fluctuating trajectory functionals, predicting the susceptibility without actually performing perturbations. Before showing examples, in the next section we describe a second procedure aimed at computing the response in a more direct way. The latter will then be compared with the fluctuation-response results above.

Iii Reweighting

In the analysis via the fluctuation-response relation exposed in the previous section, we deal with experimental data collected in steady states at various temperatures . Next we show that the same data can be used to extract a form of the susceptibility that is equivalent to Eq. (9). This means that we can bypass once again the step of the actual perturbation of the system in the laboratory.

In the definition (9) what is not useful is that one average is over trajectories under the perturbed , while the other is over unperturbed trajectories. In steady state experiments, trajectories of the former kind are not available. To sidestep this, we find it convenient to consider the alternative formula

(17)

because with this form, we can re-express both averages above in terms of steady state averages at , by the following arguments.

First, take the steady state average

(18)
(19)

Second, by denoting the probability measure of path under temperatures by [where is the path-weight, given that it starts from ], take the transient average

(20)
(21)
(22)

Thus, by this reweighting via stationary distributions, both averages appearing in Eq. (17) have been reformulated as steady state averages at , and the susceptibility becomes

(23)

The second single-time average can be written at any instant of time due to time-translation invariance. As such, substituting the particular points or one obtains, respectively,

(24)

or

(25)

Again, means the steady state average with the available data. Both formulas can be used to extract the response of the system to a step change of temperature(s) performed at . In our analysis we chose to use Eq. (24).

It is interesting to connect these expressions with previous response relations based on the knowledge of the steady state distribution. One notes that in the limit , the reweighting factor

(26)

Substituting this limit, and dropping for simplicity the temperature indices, the second expression (25) for susceptibility above becomes

(27)

implying that it comes from a response function []

(28)

Equivalently, defining the stochastic entropy ,

(29)

which is Speck and Seifert’s response formula Seifert and Speck (2010) for steady states, with the only difference that carries a physical dimension while usually the perturbation was expressed in terms of a dimensionless parameter .

While an analytical expression such as (29) is more elegant than (24) or (25), on the practical side the former may be less convenient. First of all, an analytical expression for the stationary distribution may not be known or calculable, in which case it must be actually measured at two different temperatures and the derivative will have to be performed discretely, which is equivalent to using the expressions prior to Eq. (26). Secondly, even if an analytical expression for the stationary distribution is available (as it is for the present system of interest; details in the appendix), its derivative might be too unwieldy to work with, from an implementation point of view. A discrete approximation for the derivative, such as (24), is simpler to handle. We have indeed followed this path, using analytical expressions for the distributions and , choosing with .

Iv Nonequilibrium heat capacity

In this section we show the analysis of experimental data, which show that the fluctuation-response relation with terms listed in (16), can reliably compute the susceptibility of the system. The knowledge of the steady state distribution allows us to use the formula introduced in (24), and to compute the response independently. The two versions turn out to yield results in good agreement with each other.

The averages used to compute susceptibilities are performed over trajectories that extend over ms, with time steps of length . Each trajectory is extracted by choosing a different starting point from the steady state sampling. Three cases are considered, one in the equilibrium condition  K ( trajectories) and two far from equilibrium,  K ( trajectories) and  K ( trajectories).

As an observable – where we recall that – we consider the total electrostatic energy of the system, Eq. (6), in accord with the mapping of Table 1 between electrical and mechanical quantities. The backward generator acting on this observable, appearing in (16d), becomes

(30)

where we remind that and temperatures have dimensions of energy. The response of the energy to a change of temperature becomes the nonequilibrium version of the heat capacity if (a different definition of heat capacity for nonequilibrium systems can be found in Ref. Boksenbojm et al. (2011)). The following analysis confirms that in general this heat capacity cannot be computed only from the correlation between energy and heat flowing into the system Boksenbojm et al. (2011); Baiesi et al. (2014); Yolcu and Baiesi (2016); Falasco and Baiesi (2016a, b), unless this is in equilibrium.

The susceptibility of the internal energy to a change of is shown in Fig. 2 as a function of time for the three values of . It correctly converges to a constant value for large times, though its single terms may be extensive in time in nonequilibrium conditions. We have also an analytical argument predicting that such constant value should be . It is based on recently proposed mesoscopic virial equations Falasco et al. (2015). For each degree of freedom in an overdamped system subject to multiple reservoirs we have

(31)

In our system with quadratic potential energy this implies in a steady state. Therefore, it is expected that the susceptibility as . This is indeed observed in the top panel of Fig. 2, where the steady state is an equilibrium state, with . The asymptotic value of for the susceptibility is also fairly well reached by the data in the lower panels of Fig. 2; a possible explanation of the slight disagreement is given in the next paragraph. In equilibrium (top panel), and vanish while is equal to , that is, the response is given by twice . This is essentially the Kubo formula, stating that in equilibrium the response of an observable is given by its correlations with the entropy produced in the environment (heat flow divided by reservoir temperature), which is confirmed by the form (16a) of . (The extra factor of in Eq. (16) has to do with the units of susceptibility.) On the other hand, out of equilibrium the equality between and is lost in addition to and no longer vanishing, as demonstrated by the two bottom panels of Fig. 2: All the terms and of Eq. (16) composing are all relevant. The correlation between the observable and the heat flow is not sufficient anymore in nonequilibrium systems. The frenetic terms , and the new entropy production term , are also relevant for predicting the nonequilibrium response.

K
K
K

Figure 2: (Color online) Response of the total energy to a change of , for equilibrium ( K ) and for nonequilibrium ( K and K). The susceptibility computed with the fluctuations-response relation (10) and its terms , , , are shown. The susceptibility computed with the reweighting formula (24) agrees with .

While the susceptibility at equilibrium () attains the expected asymptotic value of fairly closely, the susceptibility out of equilibrium () seems to fall a bit short. We argue that this has to do with the inevitable limitation on the time resolution of the trajectory measurements, since numerical simulations of an equivalent system also exhibit the same feature when the time discretization becomes coarse. Indeed, the sampling interval in the experiments ( ms) is not much smaller than the dynamical time scale ms in the circuit, which one can confirm visually from the plots in Fig. 2. The reason why it is the nonequilibrium susceptibilities which suffered more from this quantization error is likely as follows: Out of equilibrium, trajectory functionals like entropy production are numerically larger than in equilibrium, amplifying any error in the trajectory.

In all examples we also plot the susceptibility computed with (24). Clearly there is a very good agreement between this estimate and for all times, including the deviation from the asymptotic value for large times. This suggests that both approaches work well and corroborates our explanation of the slight offset in the asymptotic value, as also should be affected by the time-step discretization.

Conclusion

We have shown that experimental steady state data can be used to predict the thermal linear response of an electric circuit, even if it works in a thermally unbalanced nonequilibrium regime due to a cryogenic bath applied to one of the two resistors. We have used a recent nonequilibrium response relation for our analysis. This approach requires the knowledge of the forces acting on each degree of freedom, an information easily available in our case. The nonequilibrium version of the heat capacity provides a simple demonstration of the fact that in general one cannot expect to predict the response of the energy to thermal variations just from the unperturbed correlations between energy and fluctuating heat flows, as one would do by using the standard fluctuation-dissipation theorem for equilibrium systems. Also non-dissipative aspects play a crucial role: The response includes correlations between the observable and the so-called frenesy of the system Basu and Maes (2015); Baiesi and Maes (2013), which is a measure of how frantically the system wanders about in phase space. Eventually our example of generalised heat capacity should help understanding how to construct a theory for steady state thermodynamics.

In order to have a comparison with an independent method for computing the susceptibility, we also introduced a reweighting procedure that has the advantage of needing no more than the same steady state data. The second method estimates the susceptibility of the system in a more direct sense, namely mimicking actual finite perturbations of the system. This procedure is simple to implement and is related to a linear response formula also based on the knowledge of the steady state distribution.

Appendix A Gaussian steady states distributions

We review the procedure used to obtain the steady state distribution for linear overdamped stochastic systems with additive noise.

Consider a process given by the stochastic differential equation

(32)

with being -dimensional uncorrelated noise and the -dimensional state. Here, and are positive-definite constant matrices. The ensemble current corresponding to these degrees of freedom follows as

(33)

with the Fokker-Planck equation . Thus, stationarity implies (index notation hereafter)

(34)
(35)
(36)

Clearly an exponential quadratic form would satisfy this equation and the ansatz

(37)

with positive-definite, yields

(38)
(39)

Using the symmetry of and matrix notation, this can be rewritten as

(40)

Since this is supposed to hold for any , both sides must vanish. For the right-hand side, this implies that the matrix is skew-symmetric, which means that it has vanishing symmetric part,

(41)

or, equivalently,

(42)

The left-hand side of (40) also vanishes, as required, when a satisfying (42) is found.

Being a linear equation in the unknown entries of , one can imagine rewriting (42) so as to treat those unknowns as a vector (likewise the right-hand side), and afterwards inverting the matrix equation. This is achieved by resorting to the Kronecker product, denoted by , and a “vectorization” operation, denoted as “vec”, which amounts to stacking the columns of a matrix into a single column. Eq. (42) is recast in the form

(43)

Hence we find via

(44)

followed by an “un-vec”, i.e. a procedure reverting back from vectorized matrices to actual ones.

Circuit experiments

In the electric circuit experiments, the equation of motion for the charges is of the form

(45)

where

(46)
(47)
(48)
(49)

[ was defined in (4)]. Thus, we identify and . Through (44) and inverting the resulting matrix , we have

(50)

with

(51)
(52)

and

(53)
(54)
(55)
(56)

We have thus all the elements for computing the steady state distribution (37) analytically at any combination of temperatures .

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