Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems

In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function (PDF) without a need to solve the GFPE. We employ our metho...

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Datum:	2011
1. Verfasser:	Sliusarenko, O.Yu.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2011
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spelling	irk-123456789-1199762017-06-11T03:03:52Z Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems Sliusarenko, O.Yu. In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function (PDF) without a need to solve the GFPE. We employ our method to obtain the GFPE and PDFs for free generalized Brownian motion and the one in harmonic potential for the case of power-law correlation function of the noise. We prove the fact that the considered systems may be described with Einstein-Smoluchowski equation at high viscosity levels and long times. We also compare the results with those obtained by other authors. At last, we calculate PDF of thermodynamical work in the stochastic system which consists of a particle embedded in a harmonic potential moving with constant velocity, and check the work fluctuation theorem for such a system. У цiй роботiми пропонуємо послiдовний пiдхiд до виводу узагальненого рiвняння Фокера-Планка(УРФП) для Ґаусових немаркiвських процесiв iз стацiонарними прирощеннями.Цей пiдхiд дозволяє побудувати функцiю розподiлу (ФР) процесу без потреби безпосередньо розв’язувати УРФП. Ми застосовуємо цей метод для знаходження УРФП та ФР для вiльного узагальненого броунiвського руху та узагальненого броунiвського руху в потенцiалi для випадку степеневої кореляцiйної функцiї шуму. Ми доводимо,що розглянутi системи можуть описуватися у рамках рiвняння Ейнштейна-Смолуховського за умов сильної в’язкостi та великих часiв.Також ми порiвнюємо результати iз отриманими iншими авторами. Нарештi, ми обчислюємо ФР термодинамiчної роботи у стохастичнiй системi,що складається з частинки у гармонiчному потенцiалi,який рухається з постiйною швидкiстю,та перевiряємо флуктуацiйну теорему для роботи у такiй системi. 2011 Article Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems / O.Yu. Sliusarenko // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23002:1-14. — Бібліогр.: 37 назв. — англ. 1607-324X PACS: 05.10.Gg, 52.65.Ff, 02.50.Ey, 05.40.-a DOI:10.5488/CMP.14.23002 arXiv:1107.0796 http://dspace.nbuv.gov.ua/handle/123456789/119976 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function (PDF) without a need to solve the GFPE. We employ our method to obtain the GFPE and PDFs for free generalized Brownian motion and the one in harmonic potential for the case of power-law correlation function of the noise. We prove the fact that the considered systems may be described with Einstein-Smoluchowski equation at high viscosity levels and long times. We also compare the results with those obtained by other authors. At last, we calculate PDF of thermodynamical work in the stochastic system which consists of a particle embedded in a harmonic potential moving with constant velocity, and check the work fluctuation theorem for such a system.
format	Article
author	Sliusarenko, O.Yu.
spellingShingle	Sliusarenko, O.Yu. Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems Condensed Matter Physics
author_facet	Sliusarenko, O.Yu.
author_sort	Sliusarenko, O.Yu.
title	Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems
title_short	Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems
title_full	Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems
title_fullStr	Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems
title_full_unstemmed	Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems
title_sort	generalized fokker-planck equation and its solution for linear non-markovian gaussian systems
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/119976
citation_txt	Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems / O.Yu. Sliusarenko // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23002:1-14. — Бібліогр.: 37 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT sliusarenkooyu generalizedfokkerplanckequationanditssolutionforlinearnonmarkoviangaussiansystems
first_indexed	2025-07-08T17:00:35Z
last_indexed	2025-07-08T17:00:35Z
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fulltext	Condensed Matter Physics, 2011, Vol. 14, No 2, 23002: 1–14 DOI: 10.5488/CMP.14.23002 http://www.icmp.lviv.ua/journal Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems O.Yu. Sliusarenko Akhiezer Institute for Theoretical Physics NSC KIPT, 1 Akademichna Str., 61108 Kharkiv, Ukraine Received February 11, 2011 In this paper we suggest a consistent approach to derivation of generalized Fokker-Planck equation (GFPE) for Gaussian non-Markovian processes with stationary increments. This approach allows us to construct the probability density function (PDF) without a need to solve the GFPE. We employ our method to obtain the GFPE and PDFs for free generalized Brownian motion and the one in harmonic potential for the case of power-law correlation function of the noise. We prove the fact that the considered systems may be described with Einstein-Smoluchowski equation at high viscosity levels and long times. We also compare the results with those obtained by other authors. At last, we calculate PDF of thermodynamical work in the stochastic system which consists of a particle embedded in a harmonic potential moving with constant velocity, and check the work fluctuation theorem for such a system. Key words: Fokker-Planck equation, Gaussian system, non-Markovian system, thermodynamical work, transient fluctuation relation PACS: 05.10.Gg, 52.65.Ff, 02.50.Ey, 05.40.-a 1. Introduction The theory of Markovian Brownian motion is successfully used in describing a great variety of experiments and observations [1–4]. However, it remains an inapplicable model for the majority of natural systems where a characteristic time of thermal fluctuations is comparable to that of a Brownian particle (Gaussian non-Markovian systems), or where the processes are strongly non- Gaussian (either Markovian or non-Markovian), all of which results in the fact that the long-time mean squared displacement does not grow linearly in time any more, 〈 x2(t) 〉 ∝ tµ. This phe- nomenon is called anomalous diffusion, namely, when µ < 1, the system is said to be subdiffusive, and when µ > 1, it is superdiffusive. Evidently, when µ = 1 we have an ordinary Brownian motion. There are two paradigmatic models describing anomalous diffusion: continuous time random walk (CTRW) and fractional Brownian motion (FBM). The CTRW approach was developed by Montroll and Weiss in 1965 [5] for a description of the electric charge transport in a disordered medium (amorphous semiconductor) [6]. This model considers the independent identically dis- tributed couples of random space-time steps whose PDFs belong to the domain of attraction of Lévy stable laws. Recently, the Markovian Lévy processes in external fields were studied by means of Langevin and fractional kinetics technique [7–9]. The second model (FBM) was introduced by Kolmogorov in 1940 [10] and later studied by Yaglom [11]. The name “fractional Brownian motion” belongs to Mandelbrot and van Ness who suggested a stochastic integral representation of this process [12]. FBM is a continuous centered non-Markovian Gaussian process X(H)(t) with covariance function 〈 X(H)(t)X(H)(t′) 〉 = D ( t2H + t′2H − \|t− t′\|2H ) , (1.1) or, at large times, 〈 X(H)(t)2 〉 = 2Dt2H , (1.2) c© O.Yu. Sliusarenko, 2011 23002-1 http://dx.doi.org/10.5488/CMP.14.23002 http://www.icmp.lviv.ua/journal O.Yu. Sliusarenko where H is Hurst index, 0 < H < 1, and D is the generalized diffusion coefficient of the dimen- sion [D] = cm2/sec2H. Previously, the problem of particle escape from the potential well in the framework of this model was considered in paper [13] by using the method of numerical simulation of Langevin equation with fractional Gaussian noise Y (H)(t). The latter is a non-Markovian sta- tionary random process which is defined as the time derivative of FBM and whose autocorrelation function exhibits a slow decay at infinity as 〈 Y (H)(t)Y (H)(0) 〉 ≈ 2DH(2H−1)t2H−2, in contrast to white Gaussian noise, where 〈 Y (1/2)(t)Y (1/2)(t′) 〉 = 2Dδ(t− t′). Power spectral density for white noise does not depend on frequency, otherwise the noise is called a coloured noise. The pioneer work of deriving a differential equation (in essence a Fokker-Planck equation, FPE) describing ordinary Brownian motion (OBM) was done by Lord Rayleigh [14], within the approach of an absence of external potential and an overdamped discrete motion of a heavy Brownian particle. A more consistent method was developed by Fokker, Smoluchowski and Planck (a detailed historical sketch may be found in [4]). However, when dealing with the coloured noise case, the above-mentioned approaches are no longer valid. The most common example of derivation of one-dimensional Fokker-Planck equation for colou- red noise may be found in paper [4]; for one-dimensional case it was done in [15]; for a particular case of a linear oscillator it was obtained and studied in [16]. The multi-dimensional case was considered in [17]. The theory of generalized Brownian motion (GBM) finds its applications in many problems of modern physics, biophysics and astronomy. Indeed, polymers [18–20], elastic chains and mem- branes [19, 21–24] and rough surfaces [25–27] can be described by a continuum elastic model which accounts for their general stochastic behavior; it was recently shown that the probe particle in such systems performs FBM [28, 29]. The fluctuations of magnetic field in the turbulent plasma of the Earth’s magnetospheric tail turn out to have colour: in the range of frequencies ω 6 10−2 Hz they have the properties of flicker-noise (their power spectrum is proportional to 1/ω). When ω is about 10−1 Hz, they are a brown noise with the tendency of “blackening” at lower frequencies, see, e.g., the paper [30] and works cited therein. Moreover, a similar situation is known from experiments in laboratory plasmas: it was found that the power spectra of the saturation current, electrostatic potential fluctuations, and the turbulence-induced flux measured in various plasma devices [31] have power-law dependencies. At high frequencies, an asymptotic power fall-off of the fluctuation spectra with characteristic decay indices close to 2 was denoted; at intermediate frequencies, the decay indices were about 1, gaining a weak frequency dependence at the lowest frequencies. Another important application comes from single-molecule dynamics. In paper [32] it is shown, that the experimental data of the distance fluctuations between the two components of fluorescein- tyrosine complex can be described within the framework of the Langevin equation with harmonic potential and coloured source possessing correlation function (CF), which decays as t−0.51±0.07. Below we present a consistent method of derivation of a multi-dimensional generalized Fokker- Planck equation for linear stochastic systems driven by coloured Gaussian noise paying special attention to the case of coloured Gaussian noise with power-law correlation function. 2. Basics of the method We use the approach to obtaining an ordinary Fokker-Planck equation for linear systems with delta-correlated noise described in monograph [33] as the basis of the suggested method for deriva- tion of the generalized Fokker-Planck equation. The paper continues and extends the previous studies [34] where we considered the GFPE for exponential and power-law correlation function restricting ourselves only to space-homogeneous case. Here we study a more general problem for the power-law correlation function. For the integrity and clarity of presentation, we give a full description of the method, as well. First, let us write Langevin equations in multi-dimensional form: ξ̇i = −aikξk + Yi (t) +Ki . (2.1) Here ξi is the generalized coordinate, aik is the coefficient matrix, Yi is the external noise, Ki is 23002-2 Fokker-Planck equation for non-Markovian Gaussian systems the regular constant force; the dot above ξi stands for time derivative. Let the initial conditions be ξi (t = 0) = ξi (0). Then, the formal solution of (2.1) is ξi (t; ξ (0)) = ( e−at ) ij ξj (0) + t ∫ 0 dτ ( e−a(t−τ) ) ij (Yj (τ) +Kj) , (2.2) where a ≡ ‖aik‖ is matrix composed from the elements of aik . The probability density function (PDF) of the value ξi in the moment of time t with the fixed ξ (0) is evidently a multi-dimensional Dirac delta-function: f (ξ, t; ξ (0)) = δ (ξ − ξ (t, ξ (0))) ≡ ∏ i δ (ξi − ξi (t, ξ (0))) . (2.3) In case we do not know the exact ξ (0), but their initial PDF f (ξ (0) , 0), the PDF at an arbitrary moment of time t will be of the shape: f (ξ, t) = ∫ dξ (0) f (ξ (0) , 0) 〈δ (ξ − ξ (t, ξ (0)))〉 , (2.4) where 〈. . .〉 stands for ∫ dτpY (τ) . . . and pY (τ) is the PDF of noise. By using the n-dimensional delta-function representation δ (ξ) = (2π) −n ∫ dq exp (iqξ) and taking into account (2.2), we have: 〈δ (ξ − ξ (t, ξ (0)))〉 = (2π)−n ∫ dqĜ (q, t) exp [ iq ( ξ − e−atξ (0) )] , (2.5) where Ĝ (q, t) = 〈 exp { −iq t ∫ 0 dτe−a(t−τ) (Y (τ) +K) }〉 . (2.6) Here we should remark that we use a matrix notation and the hat indicates that the value is a Fourier image. Expanding the PDF into Fourier integral f (ξ, t) = (2π) −n ∫ dqeiqξ f̂ (q, t) , (2.7) we get from (2.4) and (2.5): f̂ (q, t) = Ĝ (q, t) f̂ ( ( e−at )T q, 0 ) , (2.8) where (e−at) T is a matrix transposed to e−at. Hereinafter we assume the random process Yi (t) to be a stationary Gaussian process, so that the following relations are true: 〈 Yi1 (t1) . . . Yi2n+1 (t2n+1) 〉 = 0, 〈Yi1 (t1) . . . Yi2n (t2n)〉 = ∑ gi1i2 (t1 − t2) . . . gi2n−1i2n (t2n−1 − t2n) , (2.9) where the summation is executed by all possible pair compositions of i1 , t1 ; i2 , t2 ; . . . ; i2n , t2n . The number of such pairs is (2n− 1)!! = 2n!/n!2n. gi1i2 (t1 − t2) is a certain function of time difference. Using the exponential function series expansion for (2.6) and keeping in mind (2.9) we have: Ĝ (q, t) = ∞ ∑ n=0 (−i) 2n (2n)! (2n)! n!2n   t ∫ 0 dt1 t ∫ 0 dt2qi ( e−at1 ) ij qm ( e−at1 ) ml gjl (t1 − t2)   n × exp  −iqi t ∫ 0 dτ ( e−aτ ) ij Kj   . (2.10) 23002-3 O.Yu. Sliusarenko Here and below for simplicity we write gjl (t1 − t2) instead of gj1l2 (t1 − t2). Introducing the value Mim (t) = 1 2 t ∫ 0 dt1 t ∫ 0 dt2 ( e−at1 ) ij ( e−at2 ) ml gjl (t1 − t2) , (2.11) we get for G (q, t): Ĝ (q, t) = exp  −qiqmMim (t)− iqi t ∫ 0 dτ ( e−aτ ) ij Kj   . (2.12) 2.1. Fokker-Planck equation It may be easily proven that the value Ĝ (q, t) obeys the following relation: ∂Ĝ ∂t + qiaik ∂Ĝ ∂qk + iKiqiĜ = −qiqmDim (t) Ĝ (q, t) , (2.13) where Dim (t) = dMim dt + aikMkm (t) + amkMik (t) . (2.14) On the other hand, due to an obvious equality ∂ ∂t f ( ( e−at )T q, 0 ) = −qa ∂ ∂q f ( ( e−at )T q, 0 ) (2.15) and (2.8) we can conclude that the function f (q, t) obeys the same equation as (2.13) which after the inverse Fourier transform yields: ∂f (ξ, t) ∂t +Ki ∂f (ξ, t) ∂ξi = ∂ ∂ξi [aimξmf (ξ, t)] +Dim (t) ∂2f (ξ, t) ∂ξi∂ξm , (2.16) actually being the generalized Fokker-Planck equation (GFPE). Now, let us simplify the expressions (2.14). Noticing that aik ( e−at ) kj = ∂ ∂t ( e−at ) ij , ∂gjl (t1 − t2) ∂t1 = −∂gjl (t1 − t2) ∂t2 and dMim dt = 1 2 t ∫ 0 dt1 ( e−at1 ) ij ( e−at ) ml gjl (t1 − t) + 1 2 t ∫ 0 dt2 ( e−at ) ij ( e−at2 ) ml gjl (t− t2) , we arrive at the expression Dim (t) = 1 2 t ∫ 0 dt1 [ ( e−at1 ) ij gjm (\|t1\|) + ( e−at1 ) mj gij (\|t1\|) ] . (2.17) 23002-4 Fokker-Planck equation for non-Markovian Gaussian systems 2.2. Probability density function The advantage of the described method is that there is no need in solving the Fokker-Planck equation to obtain the probability density function since we have constructed it implicitly at the stage of the GFPE derivation. Indeed, according to (2.8), knowing the Fourier image of the initial PDF f̂ (q, 0) ≡ F {f (ξ, 0)} we can easily get the PDF for an arbitrary moment of time: f (ξ, t) = F−1 { f̂ (q, t) } = F−1 { Ĝ (q, t) f̂ ( ( e−at )T q, 0 )} , (2.18) where Ĝ (q, t) is given with (2.12). However, the expressions (2.11) may be rather complicated for direct calculations regarding, e.g., a power-law correlation function g (t1 − t2). By means of integration variables change we get a much more usable expression, because now the internal integral does not contain the correlation function: Mim (t) = 1 2 t ∫ 0 dτgjl (τ) t−τ ∫ 0 dT { ( e−a(T+τ) ) ij ( e−aT ) ml + ( e−aT ) ij ( e−a(T+τ) ) ml } . 3. Generalized Brownian motion Let us now apply the derived formulae to the specific stochastic system: the generalization of the classical Brownian motion with the external random force is a stationary Gaussian noise with long memory effects. 3.1. Free generalized Brownian motion. Spatially homogenous case First, we investigate a simple system described with the following Langevin equations: dx dt = v, dv dt = −γv + Y (t) , (3.1) where x(t) is particles coordinate, γ is friction constant, Y (t) is Gaussian external noise with 〈Y (t)Y (t′)〉 ≡ g(t− t′). The coefficient matrix aim is a = [ 0 −1 0 γ ] , and gij (t1 − t2) = δi2δj2g (t1 − t2) . The solution of the homogenous system (3.1) yields: x(t) = x0 + 1− e−tγ γ v0 , v(t) = v0e −tγ . (3.2) Comparing these expressions with (2.2) we get e−at =   1 1− e−tγ γ 0 e−tγ   . (3.3) In what follows we restrict ourselves to the power-law noise correlation function of the form g(τ) = c \|τ \|βΓ(1− β) , (3.4) 23002-5 O.Yu. Sliusarenko with 0 < β < 1, which is actually the asymptotics of the CF for fractional Gaussian noise. Note, that at β → 1, we get the delta-function limit g(τ) → cδ(τ) [35]. Now, we can write out the exact values for the coefficients Dij and Mij , see equations (2.17) and (2.19), respectively: D11(t) = 0, (3.5) D12(t) = D21(t) = ct1−β 2γΓ(2− β) [1 + (1− β)Eβ(tγ)]− 1 2 cγβ−2 , (3.6) D22(t) = cγβ−1 − ct1−βEβ(tγ) Γ(1− β) , (3.7) M11(t) = −c (1− e−tγ) t1−β γ3Γ(2− β) + ct2−β γ2Γ(3− β) − 1 2 cγβ−4 + c ( −e−2tγ + 2e−tγ ) M(1 − β, 2− β, tγ)t1−β 2γ3Γ(2− β) + cEβ(tγ)t 1−β 2γ3Γ(1− β) , (3.8) M12(t) = c (1− e−tγ) t1−β 2γ2Γ(2− β) − c ( −e−2tγ + e−tγ ) t1−βM(1− β, 2 − β, tγ) 2γ2Γ(2− β) , (3.9) M22(t) = 1 2 cγβ−2 − ct1−βEβ(tγ) 2γΓ(1− β) − ce−2tγt1−βM(1− β, 2− β, tγ) 2γΓ(2− β) . (3.10) Here Eβ(t) is an integral exponential function Eβ(t) = ∞ ∫ 1 dp e−tp pβ and M(a, b, t) is Kummer’s confluent hypergeometric function: M(a, b, t) = Γ(b) Γ(a)Γ(b − a) 1 ∫ 0 duetuua−1(1 − u)b−a−1, (see, e.g. [36]). The generalized Fokker-Planck equation (2.16) for this case yields: ∂f(v, t) ∂t = ∂ ∂v [γvf(v, t)] +D22(t) ∂2f(v, t) ∂v2 . (3.11) When γt≫ 1, the latter expression takes the form ∂f(v, t) ∂t = ∂ ∂v [γvf(v, t)] + c γ ( γβ − e−tγt−β Γ(1− β) ) ∂2f(v, t) ∂v2 . (3.12) According to the procedure described in section 2.2, the PDF f(v, t) with the initial condition f(v, 0) = nδ(v − v0) reads f(v, t) = n 2 √ πσ exp [ − (v − e−tγv0) 2 4σ ] , (3.13) where σ = 1 2 cγβ−2 − ct1−βEβ(tγ) 2γΓ(1− β) − ce−2tγt1−βM(1− β, 2 − β, tγ) 2γΓ(2− β) . (3.14) 23002-6 Fokker-Planck equation for non-Markovian Gaussian systems 3.2. Free generalized Brownian motion. Spatially inhomogenous case Now we examine the same system but with inhomogenous initial condition f(x, v, 0) = nδ(x− x0)δ(v − v0). All the values for Dij(t) and Mij(t) clearly, remain the same as in the previous section, but the GFPE and the PDF do change: ∂f ∂t = −v ∂f ∂x + γ ∂(vf) ∂v + 2D12 ∂2f ∂x∂v +D22 ∂2f ∂v2 . (3.15) Again we construct the solution with the procedure explained in section 2.2: f(x, v, t\|ω = 0) = n 4π √ σ exp [ − 1 4σ M11(t) ( p M12(t) M11(t) − v0e −tγ + v )2 − p2 4M11(t) ] , (3.16) where p = v0 γ ( 1− e−tγ ) − x+ x0 (3.17) and σ = M11(t)M22(t)−M12(t) 2. (3.18) 3.3. Generalized Brownian motion of linear oscillator Here we study the most general system, though restricting ourselves to the case of a harmonic potential U (x) = ω2x2/2 . The pair of Langevin equations now have the following form: dx dt = v, dv dt = −γv − ω2x+ Y (t) , (3.19) where x(t) is particles coordinate, γ is friction constant, ω is frequency of the linear oscillator, Y (t) is the external noise. The coefficient matrix introduced in equation (2.1) is a = [ 0 −1 ω2 γ ] . Again, gij (t1 − t2) = δi2δj2g (t1 − t2) . The solution of the homogenous system (3.19) yields v (t) = A1e −γt/2eΩt/2 +A2e −γt/2e−Ωt/2, x (t) = − 1 ω2 [v̇ (t) + γv (t)] = − 1 2ω2 [ A1e −γt/2eΩt/2 (γ +Ω) +A2e −γt/2e−Ωt/2 (γ − Ω) ] , (3.20) where A1 and A2 are constants depending on the initial conditions and here we introduce the value Ω ≡ + √ γ2 − 4ω2. Assigning x (0) = x0 and v (0) = v0 , we get A1 = −2ω2x0 + v0 (γ − Ω) 2Ω , A2 = 2ω2x0 + v0 (γ +Ω) 2Ω . (3.21) Now, substituting the latter expressions into equations (3.20) and comparing the result with the formal solution (2.2) without the integral term (since we are looking for the solution of the homogenous system), we find e−at = e−γt/2     cosh ( Ωt 2 ) + γ Ω sinh ( Ωt 2 ) 2 Ω sinh ( Ωt 2 ) −2ω2 Ω sinh ( Ωt 2 ) cosh ( Ωt 2 ) − γ Ω sinh ( Ωt 2 )     . (3.22) 23002-7 O.Yu. Sliusarenko The final step before proceeding to the GFPE and the PDF evaluation is to obtain the exact expressions for Mij (t) and the generalized diffusion coefficients Dij (t) (i, j = 1, 2) for our power- law correlation function (3.4). A straightforward integration of equation (2.19) with (3.4) gives M11(t) = −ce −2apt ( amγ − 2etΩω2 ) M(1 − β, 2− β, apt)t 1−β 2γω2Ω2Γ(2 − β) − ce−tγ ( ape tΩγ − 2ω2 ) M(1− β, 2 − β, amt)t 1−β 2γω2Ω2Γ(2− β) + amcEβ(apt)t 1−β 2γω2ΩΓ(1− β) − apcEβ(amt)t 1−β 2γω2ΩΓ(1− β) + ( a2pa β m − aβpa 2 m ) c 2apamγω2Ω , (3.23) M12(t) = − ce−t(2ap+γ) 2Ω2Γ(2− β) t1−β [ ( e2apt − etγ ) M(1− β, 2− β, apt) + ( e2apt − et(γ+2Ω) ) M(1 − β, 2− β, amt) ] , (3.24) M22(t) = −apce −2apt ( γ − 2ametΩ ) M(1− β, 2− β, apt)t 1−β 2γΩ2Γ(2− β) − amce t(Ω−2ap) ( etΩγ − 2ap ) M(1− β, 2 − β, amt)t 1−β 2γΩ2Γ(2 − β) − apcEβ(apt)t 1−β 2γΩΓ(1− β) + amcEβ(amt)t 1−β 2γΩΓ(1− β) + ( aβp − aβm ) c 2γΩ , (3.25) where Ω = √ γ2 − 4ω2, ap = (γ +Ω)/2, am = (γ − Ω)/2. According to equation (2.16), the GFPE for such a system reads ∂f (x, v, t) ∂t = ∂ ∂x [−vf(x, v, t)] + ∂ ∂v [ (ω2x+ γv)f(x, v, t) ] +D11 ∂2f ∂x2 + (D12 +D21) ∂2f ∂x∂v +D22 ∂2f ∂v2 , (3.26) where D11 = 0, (3.27) D12 = D21 = amapc [Eβ(apt)− Eβ(amt)] t 1−β 2ω2ΩΓ(1− β) + c [ aβm(γ +Ω)− aβp (γ − Ω) ] 4ω2Ω , (3.28) D22 = c [amEβ(amt)− apEβ(apt)] t 1−β ΩΓ(1− β) + c ( aβp − aβm ) Ω . (3.29) At this stage we may compare these diffusion coefficients to that obtained in the paper by Wang and Masoliver [16]. We consider only the case of the external driving noise (see section 3.2 of the mentioned paper). To establish a connection with our GFPE and equation (W29) (here the letter “W” indicates the reference to the equation from the paper [16]), let us substitute equa- tions (W54), (W55) and (W14) into (W35). Now we see, that ψ(t) ≡ 2D12(t), φ(t) ≡ D22(t), i.e. we get a complete coincidence between our GFPE (3.26) and Wang’s GFPE (W29). 23002-8 Fokker-Planck equation for non-Markovian Gaussian systems The PDF is evaluated directly through relations (2.18) and (2.12) with Kj ≡ 0: f(x, v, t) = n 4π √ M11(t)M22(t)−M12(t)2 exp ( e−γt 4Ω2 (M11(t)M22(t)−M12(t)2) × { M11(t) [ eγt/2vΩ− v0 cosh (Ωt/2)Ω + ( 2x0ω 2 + v0γ ) sinh (Ωt/2) ]2 +M22(t) [ −eγt/2xΩ + x0 cosh (Ωt/2)Ω + (2v0 + x0γ) sinh (Ωt/2) ]2 + 2M12(t) [ eγt/2vΩ− v0 cosh (Ωt/2)Ω + ( 2x0ω 2 + v0γ ) sinh (Ωt/2) ] × [ −eγt/2xΩ+ x0 cosh (Ωt/2)Ω + (2v0 + x0γ) sinh (Ωt/2) ] } ) , (3.30) where Mij are given with equations (3.23–3.25). 4. Transition to Einstein-Smoluchowski equation Now let us prove that the system considered in section 3.3 may be described with Einstein- Smoluchowski equation at high viscosity levels and at long times. When ω/γ ≪ 1, we can neglect the time derivative of velocity and, therefore, the pair of Langevin equations (3.19) transforms into a single overdamped Langevin equation: dx dt = −ω 2 γ x(t) + 1 γ Y (t). (4.1) The GFPE for such a system, according to equations (4.1) and (2.16) has the following form: ∂f(x, t) ∂t = ∂ ∂x ( ω2 γ xf(x, t) ) +D(t) ∂2f(x, t) ∂x2 (4.2) with D(t) = cω2β−2 γβ+1 ( 1− Γ ( 1− β, tω2/γ ) Γ(1− β) ) . (4.3) Executing the same calculations as in the previous section, for the PDF we unfold: ρ(x, t) = n 2 √ πM(t) exp    − ( x− x0e −tω2/γ )2 4M(t)    , (4.4) where M(t) = 1 2 cγ−βω2β−4 − ct1−β 2γω2Γ(1− β) [ Eβ ( tω2 γ ) + e−2tω2/γ 1− β M ( 1− β, 2− β, tω2 γ ) ] . (4.5) Similar results were obtained by M. Cáceres in [15] for a stationary case [see equations (2.14) with (2.17) of the mentioned paper]. Expanding the coefficient (4.5) into a series at large (γt)’s and substituting it to the PDF (4.4) yield: ρ(x, t) ≈ nγβ/2ω2−β √ 2πc ( 1 + t−βγβω−2βe−tω2/γ Γ(1 − β) ) × exp [ −x2 ( γβω4−2β 2c + t−βγ2βω4−4βe−tω2/γ cΓ(1− β) ) + xx0γ βω4−2βe−tω2/γ c ] . (4.6) 23002-9 O.Yu. Sliusarenko Now we return to the PDF for the most general case (3.30) and also expand it into a series at ω/γ ≪ 1 ≪ γt: f(x, v, t) ≈ A exp [ −v2 ( − t −βγ2−2βe−ω2t/γ cΓ(1− β) + t−βe−tγω2−2β cΓ(1− β) + γ−β ( γ2 − 3ω2 ) 2c ) − x2 ( − t −2βe−tγγβω4−4β cΓ(1− β)2 + t−βγ2βω4−4βe−ω2t/γ cΓ(1− β) + ω4−2β ( γβ − γ−βω2β ) 2c ) − v ( ω2γ−βe−ω2t/γ(v0 + x0γ) c − v0e −tγγ2−β c ) − x ( v0e −tγγβ−1ω4−2β c − γβ−1ω4−2βe−ω2t/γ(v0 + x0γ) c ) − v0 2 ( γβ−2ω4−2βe−2ω2t/γ 2c − e−tγγβ−2ω4−2β c ) − x0 2 ( γβω4−2βe−2ω2t/γ 2c − e−tγγβ−2ω6−2β c ) − v0x0 ( γβ−1ω4−2βe−2ω2t/γ c − e−tγγβ−1ω4−2β c )] , (4.7) A ≈ 2nγω2−β 4πc ( 1 + ω−2βγβt−βe−ω2t/γ Γ(1− β) − e−tγ(tω)−2β Γ(1− β)2 ) . (4.8) Then, integrating it by v in the range of (−∞;∞) and neglecting the terms of the higher magnitude of smallness than exp { −ω2t/γ } we get: f̃(x, t) ∝ exp [ −x2 ( γβω4−2β 2c + t−βγ2βω4−4βe−ω2t/γ cΓ(1− β) ) + xx0γ βω4−2βe−tω2/γ c ] , (4.9) which fully corresponds to the PDF (4.6), and, therefore, proves the fact that the considered system at large times and strong friction may be described with Einstein-Smoluchowski equation. 5. GFPE for overdamped harmonic oscillator with constant drift As a final application example of the presented technique, let us study the PDF of the thermo- dynamical work w in the stochastic system which consists of a particle inside a harmonic potential moving with constant velocity v∗ , U = (k/2)[x−X(t)]2, X(t) = v∗t, x (t) is the particle’s coordi- nate. Our aim is to get the transient fluctuation relation for such a system, which will demonstrate large-deviation symmetry properties in the PDF, and compare it to the classical case. The work w is defined as follows: w(t) = ∫ dX ∂U ∂X = t ∫ 0 dt′ dX dt′ ∂U ∂X = −kv∗ t ∫ 0 dt′(x− v∗t ′). (5.1) Introducing y (t) = x(t)− v∗t, for the overdamped Langevin equation and the equation for the thermodynamical work w (t) we have: dy dt = − 1 τ y (t) + Y (t)− v∗ , dw dt = −kv∗y (t) , (5.2) 23002-10 Fokker-Planck equation for non-Markovian Gaussian systems where τ = mγ/k. Alternatively, if we consider the plane (y, w), the coefficient matrix a of the system will be a = [ 1/τ 0 kv∗ 0 ] . (5.3) Since y (t) = y0 exp(−t/τ), y0 = x0 is the initial position of the particle, w (t) = w0 + y0kv∗τ ( e−t/τ − 1 ) , Then the evolution matrix e−at = [ e−t/τ 0 kv∗τ ( e−t/τ − 1 ) 1 ] . (5.4) For the diffusion coefficients Dij (t) we have: D11(t) = cγ2τ1−β ( 1− Γ (1− β, t/τ) Γ(1− β) ) , (5.5) D12(t) = ckv∗γ 2t−β ( t/τ [(β − 1)Eβ (t/τ)− 1] + Γ(2− β) (t/τ)β ) 2Γ(2− β) , (5.6) D22(t) = 0. (5.7) The generalized Fokker-Planck equation in this case will have the form: ∂f (y, w, t) ∂t = (y τ + v∗ ) ∂f ∂y + kv∗y ∂f ∂w +D11 ∂2f ∂y2 + 2D12 (t) ∂2f ∂y∂w . (5.8) Now, considering an initial condition f (y, w, 0) = nδ (y − y0) δ (w − w0), when y0 = 0, w0 = 0 we get for the PDF: f (w, t) = n 2 √ π √ M22(t) exp { − [ kv2∗τ 2 ( t/τ + e−t/τ − 1 ) + w ]2 4M22(t) } , (5.9) where M22 = cm2v2∗γ 2t2−β 2Γ(3− β) { 2− e−t/τM [ 1, 3− β, t τ ] − e−t/τ ( 2− e−t/τ ) M [ 2− β, 3− β, t τ ]} , (5.10) which fully corresponds to the results obtained in [37]. After the relaxation stage has passed, at t≫ τ we find for the transient fluctuation relation: f (w, t) f (−w, t) = exp [ Γ(3− β)wtβ−1 cmγ ] . (5.11) Thus, the fluctuation relation for the system subjected to a coloured noise with the slowly decaying power-law correlation function differs from that for ordinary Brownian motion. As we stated above, the classical case limit is revealed at β → 1. 6. Conclusions In this paper we suggested a consistent method for derivation of the generalized Fokker-Planck equation for linear multidimensional Gaussian non-Markovian systems. Taking the case of the Gaussian systems with slowly decaying power-law correlations, we obtained the following results: 23002-11 O.Yu. Sliusarenko • Firstly, we constructed the solution of generalized Fokker-Planck equation, the probability density function, without solving it directly. • We derived generalized Fokker-Planck equation for free motion and constructed the proba- bility density function for spatially homogeneous and inhomogeneous cases. • For the case of the motion in a harmonic potential, the generalized Fokker-Planck equation and the probability density function were also obtained, and the results were compared to those of the other authors. • We show the equivalence in description of generalized Brownian motion in a harmonic poten- tial with generalized Fokker-Planck equation and generalized Einstein-Smoluchowski equation at high viscosity levels and at long times. • Finally, we investigated the probability density function for thermodynamical work in the stochastic system which consists of a particle inside a uniformly moving harmonic potential underlining strong differences in transient fluctuation relations for the generalized Brownian motion and the ordinary Brownian motion cases. Acknowledgements O. S. would like to thank A.V. Chechkin for the problem setting and for the discussion of the results, S.V. Peletminskii and Yu.V. Slyusarenko for helpful comments on the paper. 23002-12 Fokker-Planck equation for non-Markovian Gaussian systems References 1. Chandrasekhar S., Rev. Mod. Phys., 1943, 15, 1; doi:10.1103/RevModPhys.15.1. 2. Van Kampen N.G., Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam, 1992. 3. Klimontovich Yu.L. Statistical Physics. Nauka, Moscow, 1982 (in Russian) [Engl. transl.: Harwood Academic Publishers, New York, 1986]. 4. Hänggi P., Jung P. – In: Adv. Chem. Phys. Vol. 89, 2007, p. 239. John Wiley & Sons, Inc., Hoboken, NJ, USA; doi:10.1002/9780470141489.ch4. 5. Montroll E.W., Weiss G.H., J. Math. Phys., 1965, 6, 167; doi:10.1063/1.1704269. 6. Scher H., Montroll E.W., Phys. Rev. B, 1975, 12, 2455; doi:10.1103/PhysRevB.12.2455. 7. Chechkin A.V., Gonchar V.Yu., Klafter J., Metzler R. – In: Adv. Chem. Phys. 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Sliusarenko Узагальнене рiвняння Фокера-Планка та його розв’язок для лiнiйних немаркiвських Ґаусових систем О.Ю. Слюсаренко Iнститут теоретичної фiзики iм. О.I. Ахiєзера ННЦ ХФТI, Україна, 61108 Харкiв, вул. Академiчна, 1 У цiй роботi ми пропонуємо послiдовний пiдхiд до виводу узагальненого рiвняння Фокера-Планка (УРФП) для Ґаусових немаркiвських процесiв iз стацiонарними прирощеннями. Цей пiдхiд дозволяє побудувати функцiю розподiлу (ФР) процесу без потреби безпосередньо розв’язувати УРФП. Ми застосовуємо цей метод для знаходження УРФП та ФР для вiльного узагальненого броунiвського руху та узагальненого броунiвського руху в потенцiалi для випадку степеневої кореляцiйної функ- цiї шуму. Ми доводимо, що розглянутi системи можуть описуватися у рамках рiвняння Ейнштейна- Смолуховського за умов сильної в’язкостi та великих часiв. Також ми порiвнюємо результати iз отри- маними iншими авторами. Нарештi, ми обчислюємо ФР термодинамiчної роботи у стохастичнiй сис- темi, що складається з частинки у гармонiчному потенцiалi, який рухається з постiйною швидкiстю, та перевiряємо флуктуацiйну теорему для роботи у такiй системi. Ключовi слова: рiвняння Фокера-Планка, Ґаусова система, немаркiвська система, термодинамiчна робота, перехiдне флуктуацiйне спiввiдношення 23002-14 Introduction Basics of the method Fokker-Planck equation Probability density function Generalized Brownian motion Free generalized Brownian motion. Spatially homogenous case Free generalized Brownian motion. Spatially inhomogenous case Generalized Brownian motion of linear oscillator Transition to Einstein-Smoluchowski equation GFPE for overdamped harmonic oscillator with constant drift Conclusions

Generalized Fokker-Planck equation and its solution for linear non-Markovian Gaussian systems

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