Classical open systems with nonlinear nonlocal dissipation and state-dependent diffusion: Dynamical responses and the Jarzynski equality

Classical open systems with nonlinear nonlocal dissipation and state-dependent diffusion: Dynamical responses
and the Jarzynski equality

Hideo Hasegawa Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan
July 5, 2019

We have studied dynamical responses and the Jarzynski equality (JE) of classical open systems described by the generalized Caldeira-Leggett model with the nonlocal system-bath coupling. In the derived non-Markovian Langevin equation, the nonlinear nonlocal dissipative term and state-dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation-dissipation relation. Simulation results for harmonic oscillator systems have shown the following: (a) averaged responses of the system to applied sinusoidal and step forces significantly depend on model parameters of magnitudes of additive and multiplicative noises and the relaxation time of colored noise, although stationary marginal probability distribution functions are independent of them, (b) a combined effect of nonlinear dissipation and multiplicative colored noise induces enhanced fluctuations for an applied sinusoidal force, and (c) the JE holds for an applied ramp force independently of the model parameters with a work distribution function which is (symmetric) Gaussian and asymmetric non-Gaussian for additive and multiplicative noises, respectively. It has been shown that the non-Markovian Langevin equation in the local and over-damped limits is quite different from the widely adopted phenomenological Markovian Langevin equation subjected to multiplicative noise.

05.70.-a, 05.40.-a, 05.10.Gg

I Introduction

In the last almost half a century, many studies have been made on the Langevin model which is widely employed as a useful model for a wide range of stochastic phenomena (for a recent review, see Ref. Lindner04 ()). Dynamics of a Brownian particle subjected to potential is modeled by the Langevin equation given by


where dot and prime stand for derivatives with respect to time and argument, respectively, denotes dissipation, is zero-mean Gaussian white noise with , and expresses the strength of noise. Dissipation and diffusion terms satisfy the fluctuation-dissipation relation (FDR),


where is the Boltzmann constant and the temperature. The FDR implies that dissipation and diffusion processes originate from the same event. The simple Langevin equation given by Eq. (1) is based on the two assumptions: (a) a dissipation is local in time and (b) a diffusion depends on velocity but is independent of state . State-independent and state-dependent diffusions are commonly referred to as additive and multiplicative noises, respectively. Multiplicative noise can be phenomenologically described in a number of ways: for example, a diffusion term in Eq. (1) may be generalized as (for a review of study on multiplicative noise, see Ref.Munoz04 ())


where is a function of , and denote strengths of additive and multiplicative noises, respectively, and and are white noises. However, phenomenological diffusion terms given by Eqs. (3) and (4) have no microscopic bases and the FDR for such diffusions is not definite Sakaguchi01 (). The stationary probability distribution function (PDF), which is obtained from the Fokker-Planck equation (FPE) corresponding to the Langevin model including these diffusion terms, is generally different from the Boltzmann factor, Sakaguchi01 (); Sancho82 (); Anteneodo03 (); Hasegawa07 ().

The importance of going beyond the assumptions (a) and (b) has been recognized over many decades. The microscopic origin of additive noise has been proposed within the framework of system-bath Hamiltonians Ford65 (); Ullersma66 (); Caldeira81 (), which are known as Caldeira-Leggett (CL) type models. By using the generalized CL model including a nonlinear system-bath coupling, we may obtain the non-Markovian Langevin equation with nonlocal dissipation and multiplicative noise which preserves the FDR Lindenberg81 (); Pollak93 (). The nature of nonlinear dissipation and multiplicative noise has been recently explored with renewed interest Barik05 (); Chaudhuri06 (); Plyukhin07 (); Farias09 (). Ref. Barik05 () discusses a possibility of observing a quantum current in a system with quantum state-dependent diffusion and multiplicative noise. Dynamics in a metastable state which is nonlinearly coupled to bath driven by external noise has been studied Chaudhuri06 (). Ref. Plyukhin07 () investigates a temporal development in an average velocity of noninteracting Brownian particles in a finite system with nonlinear dissipative force. Quite recently, a detailed comparison is made between non-Markovian and Markovian Langevin equations including additive and multiplicative noises Farias09 (). It has been shown that in many cases, the Markovian (local) approximation is not a reliable description of the non-Markovian (non-local) dynamics Farias09 (). Nonlinear dissipation and multiplicative noise have been recognized as important ingredients in several fields such as mesoscopic scale systems Zaitsev09 (); Eichler11 () and ratchet problems Magnasco93 (); Julicher97 (); Reimann02 (); Porto00 ().

In the last decade, we have significant progress in experimental and theoretical understanding of nonequilibrium statistics (for reviews, see Refs. Busta05 (); Ritort07 (); Ciliberto10 ()). At the moment we have three kinds of fluctuation theorems: the Jarzynski equality (JE) Jarzynski97 (), the steady- and transient-fluctuation theorem Evans93 (); Narayan04 (); Crooks99 (), and the Crooks theorem Narayan04 (); Crooks99 (). These theorems are applicable to nonequilibrium systems driven arbitrarily far from the equilibrium state. In this paper, we pay our attention to the JE which was originally proposed for a classical isolated system and open system weakly coupled to baths Jarzynski97 (); Jarzynski97b (). Subsequently Jarzynski proved that the JE is valid for strongly coupled classical open systems Jarzynski04 (). A validity of the JE has been confirmed by various experiments for systems which may be described by damped harmonic oscillator models Liphardt02 (); Wang05 (); Douarche05 (); Douarche06 (); Joubaud07 (); Joubaud07b (). Stimulated by these experiments, many theoretical analyses have been made for harmonic oscillators with the use of the Markovian Langevin model Douarche05 (); Douarche06 (); Joubaud07 (); Joubaud07b (), the non-Markovian Langevin model Zamponi05 (); Mai07 (); Speck07 (); Ohkuma07 (), Fokker-Planck equation Chaudhury08 (), and Hamiltonian model Jarzynski06 (); Jarzynski08 (); Dhar05 (); Chakrabarti08 (); Hijar10 (); Hasegawa11b (). The validity of the JE has been examined for anharmonic oscillators Mai07 (); Saha06 () and for van der Pol and Rayleigh oscillators Hasegawa11c (). We should note that these studies have been made for the non-Markovian and/or Markovian Langevin models with additive noise. Recently the JE in the Markovian Langevin model with multiplicative white noise for Brownian particles has been discussed in Ref. Lev10 (). However, a study on the JE for the non-Markovian Langevin model with multiplicative colored noise is scanty at the moment Aron10 ().

The purpose of the present paper is twofold: (1) to make a detailed study of the non-Markovian Langevin model derived from the generalized CL model for classical open systems with nonlinear nonlocal dissipation and state-dependent diffusion and (2) to calculate responses to applied forces and examine a validity of the JE in the system. In the following Sec. II, we derive the non-Markovian Langevin equation, adopting the generalized CL model including nonlinear system-bath coupling Caldeira81 (); Lindenberg81 (); Pollak93 (); Barik05 () (Sec. IIA). The Ornstein-Uhlenbeck (OU) process of colored noise is taken into account. By using the two methods Pollak93 (); Farias09 (); Bao05 () in which new variables are introduced into the non-Markovian Langevin equation, we obtain a set of four first-order differential equations and the relevant multi-variate FPE. The local limit of the non-Markovian Langevin equation is examined (Sec. IIB). In Sec. III, we study harmonic oscillator systems, applying simulation method to the four differential equations mentioned above. We calculate the stationary marginal PDF of the system (Sec. IIIA) and its responses to applied sinusoidal and step forces (Sec. IIIB). In particular, frequency-dependent responses of the mean position of to sinusoidal force have been made in detail. We obtain enhanced fluctuations of induced by a combined effect of nonlinear dissipation and multiplicative noise. Applying a ramp force to the system, the validity of the JE has been examined (Sec. IIIC). In Sec. IV we discuss the over-damped limit of the Markovian Langevin model subjected to multiplicative noise. Sec. V is devoted to our conclusion.

Ii The system-bath model

ii.1 Non-Markovian Langevin equation

We consider a system of a classical oscillator coupled to a bath consisting of -body uncoupled oscillators described by the CL model Caldeira81 (); Barik05 (),




Here , and express Hamiltonians of the system, bath and interaction, respectively: , and denote position, momentum and potential, respectively, of the system: , , and stand for position, momentum, mass and frequency, respectively, of bath: the system couples to the bath nonlinearly through a function : expresses an applied external force. The original CL model adopts a linear system-bath coupling with in Eq. (7) which yields additive noise Caldeira81 (). By using the standard procedure, we obtain the generalized Langevin equation given by Caldeira81 (); Lindenberg81 (); Pollak93 (); Barik05 ()




where denotes the non-local kernel and stands for noise. Dissipation and diffusion terms given by Eqs. (9) and (10), respectively, satisfy the second-kind FDR,


where the bracket stands for the average over initial states of and Lindenberg81 (); Pollak93 (); Barik05 ().

We have adopted the OU process for the kernel given by


where and stand for the relaxation time and strength, respectively, of colored noise. The OU colored noise may be generated by the differential equation,


where expresses white noise with


Equations (13) and (14) lead to the PDF and correlation of colored noise given by


where .

ii.1.1 The method A

The two methods have been proposed to transform the non-Markovian Langevin equation given by Eq. (8) into multiple differential equations Pollak93 (); Farias09 (); Bao05 (). In the method A, we introduce a new variable Farias09 (); Bao05 (),


to obtain four first-order differential equations for , , and given by


From Eqs. (18)-(21), we obtain the multi-variate FPE for distribution of ,


ii.1.2 The method B

In the method B, we introduce a new variable Pollak93 (),


to obtain four first-order differential equations,




It is noted that white noises in Eqs. (26) and (27) come from the same origin.

A variable in Eq. (27) is isolated from the rest of variables in the four differential equations. From Eq. (27) we may obtain its stationary PDF, , given by Eq. (15). The FPE relevant to Eqs. (24)-(26) for is given by


The stationary PDF of Eq. (29) with is given by Pollak93 ()


which leads to stationary marginal PDFs,


Equations (18)-(21) in the method A are equivalent to Eqs. (24)-(27) in the method B. An advantage of the method A is that the local limit of is easily obtainable in Eqs. (18)-(21), while in the method B the analytical expression for the stationary PDF given by Eq. (30) may be derived. In our simulations to be reported in the following section, we have mainly employed the method A, whose results are partly checked by separate simulations using the method B.

ii.2 Markovian Langevin equation

It is worthwhile to examine the local limit of Eq. (8) with a kernel given by


which leads to the Markovian Langevin equation,


It is evident that the Markovian Langevin equation becomes a good approximation of the non-Markovian one in the limit of .

From Eq. (34), we obtain three differential equations,


The PDF for is given by Eq. (15). The relevant FPE for the PDF of is expressed by


The stationary distribution of Eq. (38) with is given by


This is consistent with the result of the non-Markovian Langevin equation given by Eq. (30).

Iii Harmonic oscillator systems

iii.1 Stationary marginal PDF

Simulations have been performed for harmonic oscillator systems where and in Eqs. (5)-(7) are given by


where stands for oscillator frequency of the system, and and denote magnitudes of multiplicative and additive noises, respectively. We have solved Eqs. (18)-(21) by using the Heun method Note1 () with a time step of 0.001 for parameters of , and otherwise noticed. Simulations have been made for , which are averaged over sets of initial states of Gaussian-distributed and with and . In all simulations, we have used the initial conditions of and .

First we show marginal PDFs of , , and for , which are evaluated by simulations of Eqs. (18)-(21) with discarding initial results at . Figures 1(a), (b), (c) and (d) show , , and , respectively, obtained for (solid curves), (dashed curves), (chain curves) and (dotted curves), where stands for the relaxation time of OU colored noise. We note that and in Figs. 1(a) and (b) are independent of the parameters of although and in Figs. 1(c) and (d) depend on them. Calculated and are in good agreement with Gaussian PDFs given by Eqs. (31) and (32). Equation (15) shows that is the Gaussian PDF whose variance depends on for fixed and . In contrast, Fig. 1(c) shows that is Gaussian PDF for additive noise but non-Gaussian PDF for multiplicative noise: the kurtosis of defined by


is , 3.0, 7.8 and 10.3 for , , and , respectively, where the bracket denotes an average over . Note that for the Gaussian distribution.

iii.2 Responses to applied forces

iii.2.1 Sinusoidal forces

Dynamical responses of harmonic oscillator systems to applied sinusoidal and step forces are studied. We first apply a sinusoidal force given by


where , and () denote its magnitude, period and frequency, respectively. A sinusoidal force given by Eq. (43) yields averaged outputs given by


Fourier-transformed outputs given by


have peaks at . Output magnitudes defined by


express frequency-dependent responses of the system.

In the case of additive noise only [], we may obtain the analytical result of with the susceptibility given by


where is the Fourier transform of the kernel, ,


From Eqs. (48)-(51), the output magnitude is given by


Equation (54) shows that at a resonance frequency () is monotonously increased with increasing .

In the case of a general yielding multiplicative noise, however, we cannot make an analytical study because the Fourier or Laplace transformation cannot be employed. Then we have to rely on numerical simulations of Eqs. (18)-(21). Figures 2(a) and (b) show time courses of for and , respectively, with and when a sinusoidal force given by Eq. (43) with and is applied. The results of and are almost the same for in Fig. 2(a). In contrast, for , of is a little different from that of in Fig. 2(b): an irregularity in the former is larger than that of the latter. The irregularity in for is gradually reduced at a larger (relevant result not shown).

Although the difference between for different values with additive and multiplicative noises is not so clear in Fig. 2, it becomes evident in the Fourier-transformed quantity of or . Figure 3(a) shows the dependence of output magnitudes of with (chain curve) and (solid curve) for . The chain curve for additive noise with has a resonance peak at the frequency of a bath oscillator (). The dashed curve expresses a theoretical result for additive noise calculated by Eq. (53), which is in good agreement with a relevant result of simulations except for where our simulation overestimates . In contrast, the solid curve in Fig. 3(a) for multiplicative noise has a -type peak at which is different from . Furthermore, a shape of the solid curve is rather peculiar and different from that of the chain curve for additive noise. Figure 3(b) shows a similar plot of for with (chain curve) and (solid curve). A broad peak at for multiplicative noise is smaller than that for additive noise.

Figures 4(a)-(d) show the Lissajous plots of versus with multiplicative noise of for , 5.0, 4.0 and 3.0 which correspond to , 1.26, 1.57 and 2.09, respectively. We note that is in phase with for but in anti-phase for , and that the transition from an in-phase to an anti-phase occurs at (). This is consistent with a peak position of for multiplicative noise shown in Fig. 3(a).

We study fluctuations of in the system defined by


It is noted that in stationary state without forces, we obtain in harmonic oscillator systems because marginal stationary PDFs are given by and regardless of values of , and . It is not the case when sinusoidal forces are applied to the system with multiplicative noise, as will be shown in the following. Figures 5(a) and (b) show time courses of in harmonic oscillator systems for and , respectively, for multiplicative noise of when sinusoidal forces with and , 10.0 and 100.0 are applied. All start from at , results for and 10.0 being shifted by four and two, respectively, for a clarity of the figure. For , we obtain for and 10.0. In contrast, for and is much increased than unity. This is more clearly seen in Figs. 6(a) and (b), where the stationary value of defined by


is plotted as a function of for additive (dashed curves) and multiplicative noise (solid curves). The solid curve in Fig. 6(a) expressing for with multiplicative noise has a peak at where discussed before. The magnitude of a peak for with multiplicative noise in Fig. 6(b) is less significant than that for in Fig. 6(a). On the other hand, dashed curves in Figs. 6(a) and 6(b) expressing for additive noise of are given by independently of and .

Solid and chain curves in Fig. 7 express the dependence of with and , respectively, for multiplicative noise. With increasing , with is much increased than that with because the former is closer to than the latter. For additive noise, we obtain (dashed curve) as mentioned above. These enhanced fluctuations arise from a combined effect of nonlinear dissipation and multiplicative colored noise.

iii.2.2 Step forces

Next we apply a step force given by


where is the starting time of an force with a magnitude of . Figure 8(a) shows time courses of with additive noise with for , 1.0 and 10.0 when step forces with are applied at . For , an oscillation induced by a step force applied at remains for a fairly long period after , whereas those for and are quickly damped out. Figure 8(b) shows similar time courses of for multiplicative noise with for , 1.0 and 10.0. Comparing Fig. 8(a) with Fig. 8(b), we notice that an oscillation with multiplicative noise is more quickly disappear than that with additive noise.

Figure 9 shows for four sets of , (0.2, 0.8), (0.5, 0.5) and (1.0, 0.0) with . With increasing a component of the multiplicative noise, the oscillation induced by an applied step force is rapidly decayed.

iii.3 The Jarzynski equality

The JE is expressed by Jarzynski97 (); Jarzynski04 ()


where stands for a work made in a system when its parameter is changed, the bracket means the average over the work distribution function (WDF), , of a work performed by a prescribed protocol, and denotes the free-energy difference between the initial and final equilibrium states [Eqs. (65) and (66)]. Equation (58) includes the second law of the thermodynamics, , where the equality holds for the reversible process. The JE in Eq. (58) may be rewritten as


When the WDF is Gaussian given by


we obtain




where and express mean and variance, respectively, of the WDF. Equation (61) is not valid when the WDF is non-Gaussian.

We apply a ramp force given by


where stands for a duration of the applied force with a magnitude ().

The free energy difference between equilibrium states with and (, constant) is given by Jarzynski04 (); Hasegawa11c ()




where denotes the system Hamiltonian with . After some manipulations (detail being given in the Appendix), we obtain


independently of , and , which becomes for and .

Simulations have been performed with the same parameters as in Secs. III A and III B, but over sets of initial states. Simulation results are presented in Figs. 10-12. Figure 10(a), 10(b) and 10(c) show WDFs for ramp forces of , 10.0 and 1.0, respectively, with four sets of parameters of (solid curves), (1, 0, 0.1) (dashed curves), (0, 1, 10.0) (chain curves), and (0, 1, 0.1) (dotted curves). We note in Fig. 10(a) that all WDFs for locate at with widths of and that WDFs for multiplicative noise are asymmetric non-Gaussian, while those for additive noise are (symmetric) Gaussian. Indeed, the kurtosis of the WDF given by


is , 3.0, 3.9 and 4.2 for , , , and , respectively, with . When is reduced to 10.0, behaviors of are changed. Figure 10(b) shows that WDFs for with different sets of show different behavior but with almost the same values of and . WDFs for multiplicative noise much departs from Gaussian distribution: the kurtosis of for is , 3.0, 7.0 and 5.6 for , , , and , respectively. When is further reduced to 1.0, we note in Fig. 10(c) that all WDFs become almost identical Gaussian () with and .

These changes in , and as a function of are shown in Figs. 11(a), 11(b) and 11(c), respectively, where marks express simulation results and curves are plotted only for a guide of the eye. With decreasing from 100.0 to 0.1, changes from to 0.0 while increases from 0.3 to 2.0. The kurtosis of for multiplicative noise has a maximum around whereas that for additive noise keeps independently of . We should note that and may show oscillations at if simulations are performed with finer meshes (see Fig. 3(a) and (b) in Ref. Hasegawa11b ()). The dependence of calculated by Eqs. (58) and(59) is shown in Fig. 11(d) where the equality: holds within conceivable numerical errors. The JE is expected to hold in our system independently of the parameters of , , and .

Figure 12 shows the temperature-dependent for (solid curve), (dashed curve) and (chain curve) when a ramp force of is applied to a system with multiplicative noise of . With increasing the temperature, a width of WDF is increased and its departure from the Gaussian distribution is more significant: the kurtosis of is , 15.0 and 13.0 for , and , respectively. The shape of non-Gaussian WDF with multiplicative noise considerably depends on the temperature.

Iv Discussion

The over-damped limit of the Markovian Langevin equation is conventionally derived with setting in Eq. (34). This is, however, not the case where dissipation and diffusion constants are state dependent as in our case. Sancho, San Miguel and Dürr Sancho82 () have developed an adiabatic elimination procedure to obtain an exact Langevin and FPEs in such a case. In order to adopt their method Sancho82 (), we rewrite Eq. (34) as




By the adiabatic elimination in Eq. (69) after Ref. Sancho82 (), the FPE in the Stratonovich interpretation is given by


where we employ the relation: derived from Eqs. (70) and (71). The corresponding Langevin equation is given by Sancho82 ()


Note that the second term of Eq. (73) does not appear when we obtain the over-damped Langevin equation by simply setting in Eq. (69). It is easy to see that the stationary distribution of Eq. (72) with is given by


In the case of and , Eqs. (70), (71) and (73) lead to the Langevin equation with additive and multiplicative noises given by


In the case of additive noise only (), Eq. (75) becomes


In the opposite case of multiplicative nose only (), we obtain


Equation (75) is quite different from a widely-adopted phenomenological Langevin model given by Sakaguchi01 (); Anteneodo03 (); Hasegawa07 ()


where and stand for magnitudes of additive and multiplicative noises, respectively, and and express zero-mean white noise with unit variance. The stationary PDF obtained from the PFE for Eq. (78) in the Stratonovich sense is given by


Equation (79) yields Gaussian or non-Gaussian PDF for , depending on and Sakaguchi01 (); Anteneodo03 (); Hasegawa07 (). The Langevin model given by Eq. (78) is one of origins leading to Tsallis’s nonextensive statistics Tsallis ().

V Conclusion

Dynamical responses and the JE of classical open systems have been studied with the use of the generalized CL model yielding the non-Markovian Langevin equation in which nonlinear dissipation term and state-dependent diffusion term satisfy the FDR [Eq. (11)]. Simulation results for harmonic oscillator systems are summarized as follows:

(i) marginal stationary PDFs for and are given by and