Asymptotic derivation of Langevin-like equation with non-Gaussian noise and its analytical solution
We asymptotically derive a non-linear Langevin-like equation with non-Gaussian white noise for a wide class of stochastic systems associated with multiple stochastic environments, by developing the expansion method in our previous paper [K. Kanazawa et al., arXiv: 1407.5267 (2014)]. We further obtain a full-order asymptotic formula of the steady distribution function in terms of a large friction coefficient for a non-Gaussian Langevin equation with an arbitrary non-linear frictional force. The first-order truncation of our formula leads to the independent-kick model and the higher-order correction terms directly correspond to the multiple-kicks effect during relaxation. We introduce a diagrammatic representation to illustrate the physical meaning of the high-order correction terms. As a demonstration, we apply our formula to a granular motor under Coulombic friction and get good agreement with our numerical simulations.
Stochastic theory has been a powerful tool to understand phenomena in various fields, such as physics Kubo , chemistry Prigogine , biophysics Biophysics , and economics Econophysics . In particular, the Langevin model with the white Gaussian noise is often used in modeling fluctuating systems Langevin . Its microscopic foundation has been understood for a system driven by a single stochastic environment in terms of microscopic theories vanKampen ; vanKampenB ; Zwanzig ; SekimotoZwanzig . For example, van Kampen’s theory vanKampen ; vanKampenB predicts that a stochastic system associated with a single environment is asymptotically described by a Gaussian model in the large system size limit (or equivalently, the small noise limit). Furthermore, the Gaussian Langevin model is sufficiently simple to be analytically solvable for a wide class of setups Gardiner . For these reasons, the Gaussian Langevin model has been accepted as a minimal model for the Brownian motion with a single environment, and has played an important role in the recent development of thermodynamics of small systems Bustamante ; Liphardt ; Trepagnier ; Blickle ; Garnier ; Ciliberto ; Sekimoto1 ; Sekimoto2 ; SekimotoB ; SeifertR1 ; SeifertR2 ; Evans ; Gallavotti ; Jarzynski ; Crooks ; Seifert ; Kurchan .
On the other hand, stochastic systems associated with multiple environments have not been fully understood. The role of multiple stochastic environments is significant for athermal systems, where both thermal and athermal fluctuations coexist because of external energy injection from the reservoirs. For example, athermal noise (e.g., avalanche Gabelli ; Zaklikiewicz or shot noise Blanter ) plays an important role as well as thermal noise in electrical circuits. In granular and biological systems, it is known that the granular noise Eshuis ; Gnoli1 ; Gnoli2 ; Gnoli3 and active noise Ben-Isaac ; Toyota , respectively, appear because of external vibration and consumption of adenosine triphosphate (ATP). These systems cannot be addressed by the conventional microscopic theories because they are coupled with multiple environments. A generalization of van Kampen’s approach toward athermal systems has recently been formulated in Ref. Kanazawa1 by considering systems associated with two different environments, i.e., thermal and athermal environments. In Ref. Kanazawa1 , it is predicted that athermal stochastic systems are universally characterized by Langevin-like equations driven by non-Gaussian noise, which is consistent with experimental reports on athermal fluctuations in electric, granular, and biological systems Gabelli ; Gnoli3 ; Ben-Isaac ; Toyota . Such non-Gaussian models are expected to be important in non-equilibrium statistical mechanics for athermal systems Kanazawa1 ; Luczka ; BauleCohen ; Kanazawa2 ; Morgado ; Kanazawa3 ; Kanazawa4 .
In this paper, we extend the formulation in Ref. Kanazawa1 to non-linear frictional systems. We asymptotically derive a non-linear Langevin-like equation with non-Gaussian noise in the small noise limit for the environments. We further obtain an analytic solution for an arbitrary non-Gaussian Langevin equation with a non-linear frictional force. We derive a full-order asymptotic formula in terms of a large frictional coefficient for the velocity distribution function (VDF), and show that the first-order approximation corresponds to the independent-kick model, which was phenomenologically introduced in Ref. Talbot . We also show that the higher-order terms directly correspond to the multiple-kicks effect during relaxation, and introduce a diagrammatic representation to illustrate the higher-order terms. As a demonstration, we address the stochastic motion of a granular motor under dry friction to verify the validity of our theory.
This paper is organized as follows: In Sec.2, we asymptotically derive the non-Gaussian Langevin equation with a non-linear friction by a small noise expansion. In Sec.3, we study the steady distribution function of the non-Gaussian Langevin equation, and derive the full-order asymptotic solution in terms of the inverse of the frictional coefficient. In Sec.4, we study a granular motor under dry friction and verify our formulation numerically. In Appendix. A, we apply our formulation to the nonequilibrium steady state of a rotor in granular and molecular gases. In Appendix. B, we derive the solution of the iterative integral equation for the Fourier representation of the distribution. In Appendix. C, we check the asymptotic tail of the Fourier representation of the distribution for the cubic friction. In Appendix. D, we check the validity of the first-order renormalized solution for the cubic friction. In Appendix. E, we show the detailed derivation of the cumulant function for the granular noise. In Appendix. F, we show the detailed derivation of the first-order formula of the steady distribution function for the granular motor.
2 Asymptotic derivation of non-Gaussian Langevin equations
Let us consider a Brownian particle moving in one dimensional space coupled with multiple environments (see Fig. 1(a)).
For simplicity, we assume that the mass of the particle is unity and that its motion obeys the Markovian dynamics characterized by a small parameter . As will be illustrated later, characterizes the amplitude of noise terms and corresponds to the inverse of the system size as in Refs. vanKampen ; vanKampenB . The dynamics of the velocity of the particle then obey the following master equation (so-called the differential Chapman-Kolmogorov equation Gardiner ):
where is probability density, is the number of stochastic environments, and is the time-independent Liouville operator originating from the th environment. Throughout this paper, we denote a stochastic variable by a variable with a hat such as . For with an integer , we assume that the th environment frequently interacts with the tracer particle and is described by a continuous force (the combination of the deterministic force and the Gaussian noise):
where is deterministic friction and is the variance of the Gaussian noise. Note that these operators have locality, which describe diffusion processes. In fact, the sample paths related to for are continuous but not differentiable almost everywhere Gardiner . For , we assume that the th environment rarely but strongly interacts with the tracer particle and is described by the Markovian jump process:
where is the transition rate from with velocity jump , the first term on the right-hand side (rhs) represents the probability inflow into , and the second term represents the probability outflow from . Note that these operators have non-locality because they describe non-local jump processes.
We assume that converges to zero for in a sufficiently rapid speed (e.g., for , where is the typical velocity scale and is the typical velocity jump scale). We then introduce the following synthesized Liouville operators (see Fig. 1(b)):
where , , and . The Liouville operators and describe continuous and discontinuous motions induced by stochastic forces, respectively (Fig. 2(a)). By introducing the white Gaussian noise satisfying and and the Poisson noise with transition rate , Eqs. (1), (4), and (5) are equivalent to the stochastic differential equation:
where the symbol takes the summation for velocity jump , and we introduce conditional transition rate . In the following, we denote the ensemble averages of stochastic quantities as . We here stress that the fluctuation terms and have correlation with the velocity of the tracer , which implies that the environmental fluctuation is not white noise but complicated stochastic force. We also note that the Poisson noise is the sum of -type spike noise terms (Fig. 2(b)) as
where are the times at which the Poisson flights happen and are characterized by the transition rate . The transition rate characterizes the typical interval between two successive Poisson flights as . We also note that the summation in Eq. (8) can be formally written as the integral form: Levy .
2.2 Derivation of non-Gaussian Langevin equations with non-linear friction terms
In this subsection, we derive non-Gaussian Langevin equations with non-linear friction terms for more general setups than those in Ref. Kanazawa1 . Non-linear frictions are ubiquitous in nature Persson2000 ; Wang2009 ; Wang2012 and are known to appear in systems such as granular Kawamura ; Olsson ; Jop2006 , biological Bormuth ; Veigel ; Jagota2011 and atomic-surface ones Urbash2004 ; Li2011 ; Weymouth2013 . We note that non-linear frictions can be discontinuous functions with respect to velocity in general (e.g., Coulombic friction), and their singular effects on stochastic properties have been interesting topics Eshuis ; Gnoli1 ; Gnoli2 ; Gnoli3 ; Kawarada ; Hayakawa ; deGenne ; Touchette2 ; Menzel ; Baule ; Talbot2 ; Touchette ; Sarracino ; Baule2 ; SanoHayakawa . Indeed, as will be shown in the next section, the distribution function can be strongly singular around the peak. We here introduce critical assumptions as follows:
(i) Small noise assumption: The noise amplitudes in and are small. In other words, their stochastic parts are scaled by a small positive constant as
where is a non-negative smooth function independent of and is a Markovian jump force whose transition rate (i.e., the Poisson jump rate with the jump amplitude on the condition ) is independent of and is a smooth function in terms of . We note that can be decomposed into the following form:
where . We here stress that is independent of , corresponding to the -independence of .
(ii) Strong deterministic friction: The friction function can be expanded as,
where is the scaled velocity and is independent of . This scaling implies that the frictional effect in is negligible compared with that in (see Fig. 3(a) as a schematics).
(iii) Stable deterministic friction: Both and are piecewise smooth functions of and , and have the single stable zero points as
where and are arbitrary positive numbers.
We note that the condition (i) is the weak-coupling condition between the system and the environment, which is necessary to truncate the environmental correlation. We also note that the scalings (10) and (11) are equivalent to
where is the scaled jump rate independent of with the scaled jump . The scaling (17) can be derived as follows: According to the scaling (11), the jump size by the discontinuous force should be scaled as to remove the -dependence. Then, the following relation holds:
which implies the scaling (17). The scaling (17) is essentially equivalent to that introduced by van Kampen vanKampen ; vanKampenB , where corresponds to the inverse of the system size. We also note two examples satisfying the assumptions (ii) and (iii): The first example is the viscous friction with an -independent parameter . The second example is Coulombic friction with an -independent parameter . We note that Coulombic friction appears for systems in contact with solid Kawamura ; Olsson ; Talbot . We also note that the sign function is defined as follows: For , . For , . For , .
We next derive the non-Gaussian Langevin equation using an asymptotic expansion in terms of . In the small noise limit , the steady distribution function converges to the -function around the stable point as , because the small noise expansion is a singular perturbation SingularPerturbation (see Fig. 3(b)). In order to solve this singular perturbation, we have to introduce an appropriate scaled variable
which enlarges the peak of the distribution , where the Langevin-like description is asymptotically valid (see Fig. 3(b)).
On the basis of the above assumptions (i), (ii) and (iii), let us derive non-linear Langevin equations. By introducing the scaled variable and the scaled distribution , the master equation (1) can be written as
where we have used Eq.(13) and
We then obtain the following reduced master equation in the limit
where we have introduced the friction function , the variance of the Gaussian noise , and the transition rate . Note that the transition rate is independent of , which implies that the environmental correlation disappears and the discontinuous stochastic force is reduced to white noise. Equation (22) is then equivalent to the non-linear non-Gaussian Langevin equation:
with the white non-Gaussian noise whose transition rate is given by . We note that the frictional effect only appears from the continuous force , not from the discontinuous force (see Fig. 3(a)).
2.3 Weak friction cases: Reduction to the Gaussian Langevin equation
We next analyze the case that the friction is weak or absent. We note that the original setup by van Kampen is the case without the continuous force: (see Fig. 4(a)). We make the following assumptions (ii’)-(iv’) instead of the above assumptions (ii)-(iii):
(ii’) Weak deterministic friction: The friction is scaled by as
where is independent of . This scaling implies that the frictional effect in is comparable with that in (see Fig. 4(b) as a schematic).
(iii’) Stable deterministic friction: The friction is zero (), or is a smooth function of which has a single stable zero point at as
(iv’) Stable jump force: The jump force is stable around . In other words, the following relations are assumed for the jump rate: Let us introduce the scaled variable and the scaled Kramers-Moyal coefficients
We assume that the Kramers-Moyal coefficients are smooth functions and the first-order coefficient has a single stable zero point as
Under the assumptions (i) and (ii’)-(iv’), we derive a Gaussian Langevin equation. According to the Kramers-Moyal expansion, we obtain
Then, the master equation (1) can be written as
We here introduce the following scaled variables:
Note that the appropriate scaled variables (30) are different from the scaled variable (19) in Sec.2.2. This difference is important because the introduction of appropriate scaled variables is the key to the singular perturbation. We then obtain the Kramers-Moyal expansion for the scaled distribution as
where we expand
with . In the limit , we obtain the Fokker-Planck equation:
where , , , and . The Fokker-Planck equation (33) is equivalent to the Gaussian Langevin equation as
where and are the independent white Gaussian noise terms satisfying and . Note that the frictional effect appears not only from the continuous force but also from the discontinuous force (see Fig. 4(b)). In other words, the emergence of the Gaussian property is equivalent to the emergence of the frictional effect from the discontinuous force.
2.4 Asymptotic connection from the non-Gaussian to the Gaussian theory
As we have shown, whether the system obeys the non-Gaussian Langevin equation (23) or the Gaussian one (34) depends on the amplitude of the frictional effect in . We here explain an asymptotic connection from the non-Gaussian Langevin equation (23) to the Gaussian one (34) in terms of the amplitude of the frictional effect. We first make the assumptions (i), (iv’), the linear friction , and the symmetric jump noise (or equivalently, ), and restrict our analysis to the following two cases:
The strong frictional case: is positive and independent of , i.e., . In this case, the assumptions (ii) and (iii) are satisfied. We then obtain
where is the white Gaussian noise, is the white non-Gaussian noise characterized by the transition rate , and . We here use the original variable as the representation.
The weak frictional case: is scaled as with a positive and -independent constant . In this case, the assumptions (ii’) and (iii’) are satisfied. We therefore obtain
where , , and and are the independent white Gaussian noise terms satisfying and . Note that we use the original variable again as the representation.
We now propose the following single equation which is valid for both cases 1 and 2:
In fact, Eq. (37) is reduced to Eqs. (35) and (36) to leading order in terms of for the cases 1 and 2, respectively. In the case 1, the second term on the rhs of Eq. (37) is negligible because the typical value of is the order of as shown in Sec. 2.2, which implies that Eq. (37) is reduced to Eq. (35) to leading order. In the case 2, Eq. (37) is reduced to Eq. (36) as follows. The Kramers-Moyal equation for Eq. (37) is given by
By introducing scaled variables , , and , we obtain
2.5 Discussion on the validity of the non-Gaussian Langevin equation
We first remark the relationship of our formulation to the central limit theorem (CLT) and the non-equilibrium steady state. We next generalize the concept of the non-linear temperature, which has been introduced in Ref. Kanazawa1 , to show the explicit criteria where the small noise expansion is valid. We also show that the small noise expansion fails to reproduce the tail of the distribution.
2.5.1 Relation to the central limit theorem
We explain the relation between the CLT and our theory. According to the CLT, the summation of independent and identically distributed (i.i.d) random variables asymptotically obeys the Gaussian distribution if all the cumulants of the i.i.d variables are finite. Because the white non-Gaussian noise belongs to the class of the i.i.d random variables, the simple summation of the white non-Gaussian noise asymptotically converges to the Gaussian noise for with the characteristic time scale111 The CLT time-scale can be estimated to be with the second and fourth order cumulants and . . If the relaxation time scale is sufficiently long (i.e., ), the system can be regarded as unchanged during time of the order and the CLT is valid. We therefore obtain the Gaussian Langevin equation (34) under the condition (ii), which is physically equivalent to . On the other hand, if the relaxation time scale is not long enough as , the CLT is no longer applicable because the system changes its state during time of the order . We then obtain the non-Gaussian Langevin equation (23) under the conditions (ii), which is physically equivalent to .
2.5.2 Relation to the non-equilibrium steady state
The non-Gaussian Langevin equation (23) describes a system far from equilibrium because the local detailed balance condition is not satisfied. To clarify this point, let us analyze the energy flux from the non-Gaussian to Gaussian bath on the basis of stochastic energetics Sekimoto1 ; Sekimoto2 ; SekimotoB ; Kanazawa2 . The heat absorbed by the Gaussian bath is defined by with the Stratonovich product Gardiner . The heat flux then flows from the non-Gaussian to the Gaussian bath: , where is the second cumulant of the non-Gaussian noise . Remarkably, the direction of heat flux is independent of (i.e., the thermal temperature). This result implies that the effective temperature of the non-Gaussian bath is much higher than that of the Gaussian bath. Indeed, high temperature difference is shown necessary between the two baths in the example of a granular rotor associated with rarefied molecular gas (see Appendix A). We note that this condition is valid for systems where the non-Gaussian athermal fluctuations appear.
2.5.3 Non-linear temperature
We here discuss the explicit criteria of the small noise assumption (i) by introducing the concept of the non-linear temperature. For simplicity, we make the assumptions (i) and (iv’), and consider the linear friction case with an -independent positive parameter . We then expand and as
where and are assumed to be non-zero. The essence of our expansion is to ignore the sub-leading terms as
where and are the typical values of and , respectively. Note that the typical value of relates to the effective temperature as
where , . Then, the condition (41) is equivalent to the low temperature condition:
where we have introduced the non-linear temperature
2.5.4 Tail of the distribution
We note that the Langevin-like description (23) is only valid for typical states of the system (i.e., ) and is invalid for rare states (i.e., ). This is because the small noise expansion is not a uniform asymptotic expansion in terms of the velocity . Indeed, for rare states , the higher-order terms in Eq. (20) are not negligible anymore. Fortunately, the probability of such rare trajectories is estimated to be extremely small, which ensures the validity of the Langevin-like description for typical trajectories. We note that the same limitation also exists for the original theory of van Kampen (i.e., the Gaussian Langevin equation is also an effective description for typical trajectories).
3 Asymptotic solutions for non-Gaussian Langevin equation with general non-linear friction
We have studied the derivation of the non-Gaussian Langevin equation (23). We next study their analytical solutions for the steady distribution function. Because the exact solution for the linear case (i.e., ) has been already obtained in Refs. Eliazar ; Kanazawa1 , we study the non-linear frictional case and derive a full-order asymptotic formula in terms of the frictional coefficient. We also show that the first-order truncation of the formula leads to the independent-kick model, which was phenomenologically introduced in Ref. Talbot . We verify in detail the validity of the first-order formula for some specific cases: Coulombic and cubic frictions. Furthermore, we introduce a diagrammatic representation for the multiple-kicks process during relaxation.
Let us consider the non-Gaussian Langevin equation with the non-linear friction (23). For simplicity, we focus on the case without the Gaussian noise . We assume that the velocity and time are nondimensionalized by the characteristic velocity of the friction function222 For example, in the case with the cubic friction , the characteristic velocity scale of the friction function is given by . and the characteristic interval of the Poisson noises, respectively. The steady distribution satisfies
We assume that Eq. (45) has a unique solution satisfying and . By introducing the Fourier representations
Equation (45) is reduced to
where we have introduced the cumulant function
Our goal is to obtain the analytic solution of the linear integral equation (48).
3.2 Asymptotic solution for strong friction
We here study the asymptotic expansion in terms of the inverse of the frictional coefficient. Let us assume that the friction function is scaled by a positive large parameter as
where a typical trajectory of the tracer is illustrated in Fig. 6.
We note that the relaxation time scale is proportional to , which implies that is physically equivalent to with the characteristic time interval of the Poisson noise (see Fig. 6). We also assume that all integrals appropriately converge in the following calculations. In the limit , the steady distribution converges to the -function around the stable point , i.e., , which is equivalent to . We then expand the Fourier representation in terms of the inverse of the friction coefficient as
where is a smooth function. We note that satisfies the following relation because of the conservation of the probability: