Entropy production and heat capacity of systems under time-dependent oscillating temperature

TL;DR

This paper uses stochastic thermodynamics to analyze entropy production and heat capacity in systems with sinusoidally varying temperature, revealing their relation to complex heat capacity components.

Contribution

It provides exact solutions for entropy production and heat capacity in systems under oscillating temperature using Fokker-Planck equations.

Findings

Entropy production is continuous due to out-of-equilibrium conditions.

Heat capacity has real and imaginary parts related to frequency.

Results connect dynamic heat capacity to complex heat capacity components.

Abstract

Using the stochastic thermodynamics, we determine the entropy production and the dynamic heat capacity of systems subject to a sinusoidally time dependent temperature, in which case the systems are permanently out of thermodynamic equilibrium inducing a continuous generation of entropy. The systems evolve in time according to a Fokker-Planck or to a Fokker-Planck-Kramers equation. Solutions of these equations, for the case of harmonic forces, are found exactly from which the heat flux, the production of entropy and the dynamic heat capacity are obtained as functions of the frequency of the temperature modulation. These last two quantities are shown to be related to the real an imaginary parts of the complex heat capacity.

Equations

$$C=\frac{-{\Phi }_{\mathrm{q}}}{dT/dt}.$$

$$\frac{dS}{dt}=\Pi -\Phi .$$

$$\Phi =\frac{{\Phi }_{\mathsf{q}}}{T},$$

$$T=T_0+T_1\cos \omega t,$$

$$\frac{\partial P}{\partial t}=-\sum _i\frac{\partial J_i}{\partial x_i},$$

$$J_i=\frac{1}{\alpha }(f_{i}P-k_{\text{mbox}B}T\frac{\partial P}{\partial x_i}),$$

$$P_0=\frac{1}{Z}{e}^{-V/k_{\text{mbox}B}T}$$

$$\frac{dU}{dt}=-{\Phi }_{\mathrm{q}},$$

$${\Phi }_{\mathrm{q}}=\frac{1}{\alpha }\sum _i(⟨f_i^2⟩+k_{\text{mbox}B}T⟨f_{ii}⟩),$$

$$C=\frac{-{\Phi }_{\mathrm{q}}}{dT/dt},$$

$$S=-k_{\text{mbox}B}\int P\ln Pdx,$$

$$\frac{dS}{dt}=\Pi -\Phi ,$$

$$\Pi =\frac{\alpha }{T}\sum _i\int \frac{J_i^2}{P}dx,$$

$$\Phi =\frac{{\Phi }_{\mathrm{q}}}{T}.$$

$$\frac{\partial P}{\partial t}=-\frac{\partial J}{\partial x},$$

$$J=-\frac{k}{\alpha }xP-\frac{k_{\text{mbox}B}T}{\alpha }\frac{\partial P}{\partial x}.$$

$$P=\frac{1}{\zeta }\exp \{-\frac{1}{2}b{x}^2\},$$

$$\alpha \frac{d}{dt}B=-2kB+2k_{\text{mbox}B}T.$$

$$T=T_0+T_1\cos \omega t,$$

$$B=\frac{k_{\text{mbox}B}{T}_0}{k}+2k_{\text{mbox}B}{T}_1\frac{2k\cos \omega t+\alpha \omega \sin \omega t}{{\alpha }^2{\omega }^2+4k^2}.$$

$${\Phi }_{\mathrm{q}}=\frac{k}{\alpha }(kB-k_{\text{mbox}B}T),$$

$${\Phi }_{\mathrm{q}}=k_{\text{mbox}B}{T}_1\omega k\frac{2k\sin \omega t-\alpha \omega \cos \omega t}{{\alpha }^2{\omega }^2+4k^2}.$$

$$\overline{\Pi }=k_{\text{mbox}B}\lambda \frac{\alpha {\omega }^{2}k}{{\alpha }^2{\omega }^2+4k^2},$$

$$\lambda =\frac{T_0}{\sqrt{T_0^2-T_1^2}}-1,$$

$$\overline{C}=k_{\text{mbox}B}\frac{2k^2}{{\alpha }^2{\omega }^2+4k^2}.$$

$$\frac{\partial P}{\partial t}=-\frac{\partial J}{\partial v},$$

$$J=-\gamma vP-\frac{\gamma k_{\text{mbox}B}T}{m}\frac{\partial P}{\partial v},$$

$${\Phi }_{\mathrm{q}}=k_{\text{mbox}B}{T}_1\omega \gamma \frac{2\gamma \sin \omega t-\omega \cos \omega t}{{\omega }^2+4{\gamma }^2}.$$

$$\overline{\Pi }=k_{\text{mbox}B}\lambda \frac{\gamma {\omega }^2}{{\omega }^2+4{\gamma }^2},$$

$$\overline{C}=k_{\text{mbox}B}\frac{2{\gamma }^2}{{\omega }^2+4{\gamma }^2}.$$

Full text

I Introduction

The investigation of systems under time dependent fields of various types is very common in experimental physics. Less common is the investigation of systems under time dependent temperature. Nevertheless, temperature oscillations is the basis of modulation calorimetry hohne2003 ; kraftmakher2004 ; filippov1966 ; sullivan1968 ; gobrecht1971 ; birge1985 ; gill1993 ; schawe1995 ; jeong1997 ; schawe1997 ; hohne1997 ; simon1997 ; jones1997 ; baur1998 ; claudy2000 ; garden2007a ; garden2007c ; garden2008 , which allows the experimental determination of the heat capacity. The method consists in heating a sample by a periodical heating power with an angular frequency $\omega$ and measuring the temperature oscillations. This procedure induces a flow of heat from which the dynamic heat capacity $C$ can be obtained as the ratio between the heat flux $\Phi_{\rm q}$ and the time variation of the temperature,

[TABLE]

The heat flux as well as the heat capacity oscillate in time with the same frequency $\omega$ of the temperature oscillations but with a phase shift. During a cycle the net heat flux vanishes but not the dynamic heat capacity. Denoting by a bar the time average of a quantity, which is its integral over a cycle divided by the period of the cycle, then $\overline{\Phi}_{\sf q}=0$ and $\overline{C}$ is nonzero and shows a dispersion, that is, a dependence on $\omega$. The conventional heat capacity $C_{0}$, or static heat capacity, is obtained in the limiting value of $\overline{C}$ when $\omega\to 0$.

Under a time oscillating temperature, the system is permanently out of equilibrium causing a continuous production of entropy as well as a continuous flux of entropy. The entropy $S$ of the system also varies in time, the time variation being equal to the rate of entropy production $\Pi$ minus the entropy flux $\Phi$,

[TABLE]

According to the second law of thermodynamics, the rate of entropy production is never negative, $\Pi\geq 0$, but the flux of entropy $\Phi$, given by

[TABLE]

may have either sign. Although $\overline{\Phi}_{\sf q}=0$, this is not the case of $\overline{\Phi}$. In fact, considering that the entropy $S$ is periodic, the left-hand side of (2) vanishes in a cycle and the net flux becomes equal to the entropy produced during a cycle, that is, $\overline{\Phi}=\overline{\Pi}\geq 0$.

Our main purpose here is the calculation of the entropy production and the dynamic heat capacity for systems subject to a temperature modulation of the type

[TABLE]

where $T_{1}$ is the amplitude of modulation and $T_{0}$ is the mean temperature. Our calculation is based on stochastic thermodynamics of systems with continuous space of states tome2006 ; tome2010 ; broeck2010 ; spinney2012 ; zhang2012a ; seifert2012 ; santillan2013 ; luposchainsky2013 ; wu2014 ; tome2015 ; tome2015book . We restrict ourselves to the case of systems of particles interacting through harmonic forces, in which case the evolution equation can be solved exactly. From its solution, we determine the rate of entropy production and dynamic heat capacity as a function of the frequency $\omega$. We also show that the dynamic heat capacity and the entropy production are related to the real and imaginary parts of the complex heat capacity, respectively.

II Fokker-Planck equation

II.1 General formulation

We consider a system of interacting particles that is described by a probability distribution $P(x,t)$ of state $x$ at time $t$, where $x$ denotes the collection of particle positions $x_{i}$. We assume that the time evolution of the probability distribution is governed by the Fokker-Planck (FP) equation tome2006 ; tome2015book

[TABLE]

where

[TABLE]

and $f_{i}=-\partial V/\partial x_{i}$ is the force acting on particle $i$, $V$ being the potential energy, $\alpha$ is a constant and $k_{\mbox{\tiny B}}$ is the Boltzmann constant.

The FP equation describes the contact of the system with a heat reservoir at temperature $T$. Indeed, it is easily shown by replacement into the FP equation that the Gibbs distribution

[TABLE]

is the stationary solution when $T$ is kept constant and, in fact, the equilibrium solution.

The time variation of the energy $U=\langle V\rangle$ of the system can be obtained from the FP equation and is

[TABLE]

where $\Phi_{\rm q}$ is the heat flux from the system to outside and is expressed by tome2006 ,

[TABLE]

where $f_{ii}=\partial f_{i}/\partial x_{i}$. Once the heat flux is known, the dynamic heat capacity is determined by

[TABLE]

if $T$ is time dependent.

From the FP equation we can also determine the time variation of the entropy

[TABLE]

which can be split in two terms,

[TABLE]

where $\Pi$ is the rate of entropy production having the following form tome2006

[TABLE]

and $\Phi$ is the entropy flux from the system to the environment

[TABLE]

II.2 Harmonic forces

When the forces are harmonic it is possible to exactly solve the FP equation even for the case of a time dependent temperature. Here, we consider a collection of independent harmonic oscillators in which case it suffices to treat just one oscillator. The potential energy of the oscillator is $V=kx^{2}/2$ which yields a force $f=-kx$ and the FP equations to be solved is

[TABLE]

where

[TABLE]

The solution of the FP equation for a time dependent temperature is a Gaussian distribution

[TABLE]

where the coefficients $b$ is time dependent. That $P$ is a solution can be checked by replacing it into the FP equation (15). Instead of seeking for the coefficients $b$, we choose to find the averages $B=\langle x^{2}\rangle$. Once $B$ is found we may get $b$, if necessary, from the relation $b=1/B$.

From the FP equation, we find the equation for $B$

[TABLE]

For $T$ depending on time like

[TABLE]

the solution of equation (18) is found to be

[TABLE]

II.3 Entropy production and heat capacity

From equation (9), it follows that the heat flux is determined by

[TABLE]

or in an explicit form as

[TABLE]

The entropy flux $\Phi$ and the dynamic heat capacity $C$ are determined from $\Phi_{\rm q}$ by the use of equations (14) and (10).

We proceed now to determine the time averages of $\Phi$ and $C$. The time average of the heat flux vanishes $\overline{\Phi}_{\rm q}=0$ as expected but not $\overline{\Phi}$ and $\overline{C}$. Carrying out the integration of $\Phi$ and $C$ over a cycle, and considering that $\overline{\Phi}=\overline{\Pi}$, we find

[TABLE]

where

[TABLE]

and the dynamic heat capacity is found to be

[TABLE]

II.4 Harmonic oscillator

The approach we have used above, by employing the FP equation (5) or (15), is appropriate do describe overdamped systems. In this approach the positions were taken into account but not the velocities. However, the oscillations of temperature affect not only the positions but also the velocities of particles. The treatment of the response of the system concerning the velocities is carried out by setting up the following FP equation that gives the evolution of the probability distribution of velocites,

[TABLE]

where

[TABLE]

which describes a free particle in contact with a reservoir at a temperature $T$.

Equation (26) is formally identical to equation (15) and we may proceed in a similar way to determine the entropy production and the heat capacities. The result for the heat flux is

[TABLE]

and the time average of the rate of entropy is

[TABLE]

where $\lambda$ is given by (24), and the dynamic heat capacity is

[TABLE]

To find the entropy production of a harmonic oscillator we should add the entropy production concerning the positions, given by (23), with the entropy production concerning the velocities, given by (29). The result is

[TABLE]

Similarly, the dynamic heat capacity is the sum of (25) and (30),

[TABLE]

The quantities $\alpha$ and $\gamma$ are related to $\alpha=m\gamma$, and $k$ is related to $\kappa$ by $k=m\kappa$.

II.5 Complex heat capacity

The dispersion of the dynamic heat capacity on frequencies, induced by a time varying temperature, has an analogy with the dispersion of susceptibility on frequencies induced by a time varying field. In this last case, the response to the field oscillation is described by a complex susceptibility. Analogously, it is also possible to define a complex heat capacity to conveniently describe the response to temperature oscillations. In fact, the complex heat capacity has been the subject of investigation in relation to temperature modulation gobrecht1971 ; birge1985 ; gill1993 ; schawe1995 ; jeong1997 ; schawe1997 ; hohne1997 ; simon1997 ; jones1997 ; baur1998 ; claudy2000 ; garden2007a ; garden2007c ; garden2008

Suppose that we replace $T$ in equation (18) by

[TABLE]

Then, instead of equation (22) and (28), we would get the following expression for the heat flux of the harmonic oscillator,

[TABLE]

By analogy with (10), a complex heat capacity $C_{c}$ can be defined by

[TABLE]

from which we find

[TABLE]

which is time independent. Comparing with expressions (31) and (32), we see that

[TABLE]

These results show that the real part of the complex heat capacity is identified with the dynamic heat capacity and the imaginary part is proportional to the rate of entropy production.

The real and imaginary parts of the complex heat capacity $C_{c}$ are shown in Fig. 1 as functions of the frequency $\omega$ for several values of $\kappa$. The real part, which is the dynamic heat capacity $\overline{C}$, becomes the static heat capacity when $\omega\to 0$, which is $C_{0}=k_{\mbox{\tiny B}}/2$ if $\kappa=0$ and $C_{0}=k_{\mbox{\tiny B}}$ if $\kappa\neq 0$. In the opposite limit, $\omega\to\infty$, it vanishes as $1/\omega^{2}$. The imaginary part vanishes when $\omega\to 0$ and so does the rate of entropy production $\overline{\Pi}$. In the limit $\omega\to\infty$, the imaginary part vanishes as $1/\omega$ but the rate of entropy production reaches a finite value, which is $\overline{\Pi}=k_{\mbox{\tiny B}}\lambda(\gamma+\kappa/\gamma)$. In Fig. 2 we have plotted $\Im(C_{c})$ versus $\Re(C_{c})$ and we sees that the curves are symmetric.

III Fokker-Planck-Kramers equation

III.1 General formulation

We consider again a system consisting of several interacting particles in contact with a temperature reservoir at temperature $T$, with which it exchanges heat. The time evolution of the probability distribution $P(x,v,t)$, where $x$ denotes the collection of the positions $x_{i}$ and $v$ the collection of velocities $v_{i}$ of the particle, is governed by the Fokker-Planck-Kramers (FPK) equation tome2010 ; tome2015 ; tome2015book

[TABLE]

where

[TABLE]

Here, $m$ is the mass of each particle, $\gamma$ is the dissipation constant, and $f_{i}$ the force acting on the particle $i$, given by $f_{i}=-\partial V/\partial x_{i}$.

If the temperature $T$ is kept constant, then for large times the probability distribution approaches the Gibbs equilibrium distribution,

[TABLE]

where $E=mv^{2}/2+V$ is the energy of the system. This result shows that the FPK equation (38) indeed describes the contact of a system with a heat reservoir at a temperature $T$.

The time variation of the energy $U=\langle E\rangle$ is obtained from the FPK equation and is

[TABLE]

where the heat flux $\Phi_{\rm q}$ from the system to outside is expressed as tome2010 ; tome2015

[TABLE]

where the first and second terms are understood as the heating power and the power of heat losses, respectively, with $\gamma k_{\mbox{\tiny B}}$ being the heat transfer coefficient kraftmakher2004 .

The entropy $S$ of the system is determined from the Gibbs expression

[TABLE]

Using the FPK equation, one finds that its time derivative can be split into two terms,

[TABLE]

where the first is the rate of entropy production which can be written as tome2010 ; tome2015

[TABLE]

and the second is the flux of entropy which can be written in the following form

[TABLE]

where $\Phi_{\rm q}$ is the heat flux given by (42). If $T$ is time dependent then the dynamic heat capacity is obtained by

[TABLE]

III.2 Harmonic oscillator

We consider here the case of just one harmonic oscillator. When the temperature or the external force is time dependent, the probability distribution (40) is no longer the solution of the Fokker-Planck equation for long times, and we should seek for a solution. When the force is harmonic, which we write as $f=-m\kappa x$, the FPK equation can be solved exactly. The solution is a Gaussian distribution in $x$ and $v$ of the type

[TABLE]

where the parameters $a$, $b$, and $c$ depend on time. That this Gaussian distribution is a solution can be checked by replacing it into the FPK equation. The solution is reduced to the determination of the time dependence of the parameters.

From the Gaussian distribution (48) we see that the parameters $a$, $b$, and $c$ are related to the averages $A=\langle v^{2}\rangle$, $B=\langle x^{2}\rangle$ and $C=\langle xv\rangle$ as follows

[TABLE]

The method we use here rests on setting up equations for $A$, $B$, and $C$, from whose solutions we can find the coefficients $a$, $b$, and $c$ of the Gaussian distribution as functions of temperature, if needed.

From the FPK equations the following set of equations are found for $A$, $B$, and $C$

[TABLE]

[TABLE]

[TABLE]

Equations (50), (51), and (52) are coupled linear differential equations whose solution can also be found for a temperature modulation of the type

[TABLE]

The solution of the set of equations (50), (51), and (52) gives the following result for $A$,

[TABLE]

[TABLE]

[TABLE]

III.3 Entropy production and heat capacity

Using the result (54) for $A$, we can write the heat flux

[TABLE]

in the explicit form

[TABLE]

The entropy flux $\Phi$ and the dynamic heat capacity $C$ are obtained from this expression for $\Phi_{\rm q}$ and by the use of equations (46) and (47). To get the time averages of $\Phi$ and $C$ we should integrate them over one cycle. Carrying out the integration, and taking into account that $\overline{\Phi}=\overline{\Pi}$, we find

[TABLE]

or in a explicit form

[TABLE]

where $\lambda$ is given by equation (24), and

[TABLE]

or in a explicit form

[TABLE]

The results above were obtained for the case of a harmonic oscillator. It is possible to find the results for a free particle by formally setting $\kappa=0$. Using this procedure we recover the results (29) and (30) for a free particle.

III.4 Complex heat capacity

Again we may set up the complex heat capacity. If equations (50), (51), and (52) are solved by replacing the temperature $T$ by

[TABLE]

then instead of expression (58) we would get

[TABLE]

and the following complex heat capacity

[TABLE]

which is time independent. The real and imaginary parts of $C_{c}$ are

[TABLE]

and using relations (59) and (61) we find

[TABLE]

Again, these results show that the real part of the complex heat capacity is the dynamic heat capacity and the imaginary part is proportional to the rate of entropy production.

The real and imaginary parts of the complex heat capacity $C_{c}$ are shown in Fig. 3 as functions of the frequency $\omega$ for several values of $\kappa$. The real part, which is the dynamic heat capacity $\overline{C}$, becomes the static heat capacity when $\omega\to 0$, which is $C_{0}=k_{\mbox{\tiny B}}/2$ if $\kappa=0$ and $C_{0}=k_{\mbox{\tiny B}}$ if $\kappa\neq 0$. In the opposite limit, $\omega\to\infty$, it vanishes as $1/\omega^{2}$. The imaginary part vanishes when $\omega\to 0$ and so does the rate of entropy production $\overline{\Pi}$. In the limit $\omega\to\infty$, the imaginary part vanishes as $1/\omega$ but the rate of entropy production reaches a finite value, which is $\overline{\Pi}=k_{\mbox{\tiny B}}\lambda\gamma$. In Fig. 4 we have plotted $\Im(C_{c})$ versus $\Re(C_{c})$.

III.5 Dynamic heat capacity

During a small interval of time $\Delta t$, the heat introduced equals $-\Phi_{\rm q}\Delta t$, which divided by the increment $\Delta T$ in temperature gives $\Phi_{\rm q}\Delta t/\Delta T$. The heat capacity is obtained by taking the limit $\Delta t\to 0$,

[TABLE]

which is the expression of the dynamic heat capacity that we have used. In the absence of external work, which is the case of the present analysis, $-\Phi_{\rm q}=dU/dt$ and the heat capacity is related to the energy by $C=(dU/dt)/(dT/dt)$.

The dynamic heat capacity does not share with the static heat capacity $C_{0}$ the property $C_{0}\geq 0$. Generically, the heat flux is not in phase with the variation of temperature. A flux of heat to outside could happen while the temperature is increasing, or a flux toward the system could happen while the temperature is decreasing. In both cases the dynamic heat capacity has a negative sign. This peculiar but not illegitimate behavior is shown in Fig. 3a for a small interval of frequencies for one of the curves. Notice, on the other hand, that the rate of entropy production is always nonnegative as illustrated in Fig. 3b.

IV Conclusion

We have determined the entropy production and the dynamic heat capacity of systems under time varying temperature by the use of stochastic thermodynamics. The systems that we have analyzed evolves in time according to the Fokker-Planck, for the overdamped case, or to the Fokker-Planck-Kramers equation. Exact solutions were possible to find for the cases of harmonic forces and temperature modulation of the sinusoidal type. The heat flux also varies sinusoidally but with a phase shift with respect to temperature. From the heat flux, the rate of entropy production $\overline{\Pi}$ and the dynamic heat capacity $\overline{C}$ could be determined as functions of the frequency $\omega$ of the temperature modulation. In the limit of small frequencies, $\overline{C}$ approaches the equilibrium heat capacity, which is nonnegative, and vanishes for large frequencies. The dynamic heat capacity may not be a monotonic decreasing function of $\omega$ and might even be negative. The rate of entropy production is always nonnegative, vanishing for zero frequency, when the system is in equilibrium. For large values of $\omega$ it approaches a nonzero value. Finally, $\overline{C}$ and $\overline{\Pi}$ were shown to be related to the real an imaginary parts of the complex heat capacity.

The calculation of the production of entropy and heat capacity that we made above can be extended to a harmonic solid whose potential energy is

[TABLE]

where $K_{ij}$ are the elements of a symmetric matrix with nonnegative eigenvalues $m\kappa_{\ell}$. The harmonic solid can be understood as a collection of independent harmonic oscillators with distinct frequencies that are $\sqrt{\kappa_{\ell}}$. Therefore, we expect the total entropy production and the total heat capacity to be the sum of the entropy productions and the heat capacities of the individual oscillator, each one having a spring constant equal to $m\kappa_{\ell}$,

[TABLE]

where $\overline{\Pi}(\kappa)$ and $\overline{C}(\kappa)$ are given by the expressions (60) and (62), or in the overdamped case by the expressions (60) and (62), respectively.