Comment on "Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass"
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Comment on ”Deformed Fokker-Planck equation: inhomogeneous medium with aposition-dependent mass”
J. Luczka
Institute of Physics, University of Silesia, 41-500 Chorz´ow, Poland ∗ In a recent paper by B. G. da Costa et al. [Phys. Rev. E 102, 062105(2020)], the phenomenologi-cal Langevin equation and the corresponding Fokker-Planck equation for an inhomogeneous mediumwith a position-dependent particle mass and position-dependent damping coefficient have been stud-ied. The aim of this comment is to present a microscopic derivation of the Langevin equation forsuch a system. It is not equivalent to that in the commented paper.
In Ref. [1] the authors write that the goal of the paper is to present the Fokker-Planck equation for an inhomogeneousmedium with a variable diffusion coefficient within the position-dependent mass scenario. They consider a particleof mass m ( x ) in a fluid of viscosity coefficient λ ( x ) subjected to an external potential force F ( x ) and a random force R ( t ). They propose the Langevin equation of the form (see Eq. (46) therein) m ( x )¨ x + 12 m ′ ( x ) ˙ x = − m ( x ) λ ( x ) ˙ x + F ( x ) + R ( t ) , (1)where the dot and the prime denotes a differentiation with respect to time and the coordinate x , respectively. Next,they write that in the overdamped limit ( λ ( x ) >> τ − with τ a coarse-grained timescale) the left-hand side of Eq.(1) vanishes and then it reduces to the form˙ x = 1 m ( x ) λ ( x ) [ F ( x ) + R ( t )] . (2)The authors do not define in a precise way quantities and parameters of the model. In consequence, the reader isconfused. If m ( x ) = m = const. and λ ( x ) = λ = const. then one can rescale parameters to obtain a consistentdescription. E.g for the Stokes frictional force, the first term on the right-hand side of Eq. (1) is presented in astandard form − γ ˙ x , where γ is the friction coefficient and for consistency one should assume that λ = γ /m .Then one obtains a correct equilibrium state. However, in the case of x -dependence, from Eq. (2) it follows thatin the stationary state the corresponding probability distribution P s ( x ) for the particle position depends on its mass m ( x ) which is not correct. The authors cite the van Kampen paper on diffusion in inhomogeneous media [2], however P s ( x ) therein does not depend on the particle mass.We want to present an alternative, more consistent and microscopic model in terms of the Caldeira-Leggett frame-work. The system consists of a Brownian particle plus thermostat of an infinite number of harmonic oscillators. TheHamilton function for such a system reads [3, 4] H = p m ( x ) + V ( x ) + X i (cid:20) p i m i + m i ω i q i − η i g ( x )) (cid:21) , (3)where the coordinate and momentum { x, p } refer to the Brownian particle of coordinate-dependent mass m = m ( x )subjected to the potential V ( x ) and { q i , p i } are the coordinate and momentum of the i -th heat bath oscillator ofmass m i and the eigenfrequency ω i . The function g ( x ) couples the bath modes nonlinearly to the coordinate of theBrownian particle and the parameter η i characterizes the interaction strength of the particle with the i -th oscillator.All coordinate and momentum variables obey canonical equal-time Poisson-bracket relations.The next step is to write the Hamilton equations of motion for all coordinate and momentum variables. For theBrownian particle, the Hamilton equations read˙ x = ∂H∂p = 1 m ( x ) p, (4)˙ p = − ∂H∂x = m ′ ( x ) p m ( x ) − V ′ ( x ) + g ′ ( x ) X i c i ( q i − η i g ( x )) , (5) ∗ Electronic address: [email protected] where c i = η i m i ω i . (6)For the thermostat variables, one gets: ˙ q i = ∂H∂p i = p i m i , (7)˙ p i = − ∂H∂q i = − m i ω i q i + c i g ( x ) . (8)What we need in Eq. (5) is the solution q i = q i ( t ), which can be obtained from Eqs. (7) and (8) with the result q i ( t ) = q i (0) cos( ω i t ) + p i (0) m i ω i sin( ω i t ) + c i m i ω i Z t sin[ ω i ( t − s )] g ( x ( s )) ds. (9)The following step is to integrate by parts the last term in Eqs. (9) and insert it into Eq. (5) for p = p ( t ). UsingEq. (4), after some algebra, one can obtain an effective equation of motion for the particle coordinate x ( t ). It is calleda generalized Langevin equation and has the form m ( x ( t )) ¨ x ( t ) + 12 m ′ ( x ( t )) ˙ x ( t ) = − V ′ ( x ( t )) − g ′ ( x ( t )) Z t γ ( t − s ) g ′ ( x ( s )) ˙ x ( s ) ds − γ ( t ) g ′ ( x ( t )) g ( x (0)) + g ′ ( x ( t )) R ( t ) , t > , (10)where γ ( t ) is a dissipation function (damping or memory kernel) and R ( t ) denotes the random force, γ ( t ) = X i c i m i ω i cos( ω i t ) , (11) R ( t ) = X i c i (cid:20) q i (0) cos( ω i t ) + p i (0) m i ω i sin( ω i t ) (cid:21) . (12)In the standard approach, it is assumed that the initial probability distribution of the total system is factorized withan arbitrary probability density for the Brownian particle and the canonical Gaussian distribution for thermostat. Insuch a case, R ( t ) is a Gaussian random force of zero mean obeying the fluctuation-dissipation relation h R ( t ) R ( s ) i = k B T γ ( t − s ) , (13)where the brackets denote canonical averaging over the bath modes.The dynamics of the Brownian particle is therefore described by a stochastic integro-differential equation for thecoordinate x ( t ). In the case of a constant mass m ( x ) = m , Eq. (10) has been intensively applied in the theoryof thermally activated rate processes [4, 5]. When m ( x ) = m and g ( x ) = x it reduces to the standard generalizedLangevin equation, namely, m ¨ x ( t ) + Z t γ ( t − s ) ˙ x ( s ) ds = − V ′ ( x ( t )) − γ ( t ) x (0) + R ( t ) , t > . (14)Now, we can assume the Ohmic dissipation which means that the memory kernel is a Dirac delta, γ ( t ) = 2 γ δ ( t ),where γ >
0. In such a case, Eq. (10) takes the form m ( x ) ¨ x + 12 m ′ ( x ) ˙ x = − γ [ g ′ ( x )] ˙ x + F ( x ) + g ′ ( x ) R ( t ) , x = x ( t ) , F ( x ) = − V ′ ( x ) , t > . (15)This equation is not equivalent to Eq. (1). First, Eq. (15) is a stochastic equation with multiplicative noise. Secondly,the deterministic part is different. In particular, the damping term does not depend on the particle mass. The authorsof Ref. [1] neglect two terms in the left-hand side of Eq. (1) (calling this procedure as an overdamped limit) andanalyse the corresponding Fokker-Planck equation which depends on mass m ( x ) via the first term in the right-handside of Eq. (1). If one apply the same procedure to Eq. (15) then the corresponding Fokker-Planck equation takesa different form and does not depend on mass m ( x ) as it should be. Finally, we want to stress that the overdampedlimit has to be precisely defined by rescaling Eq. (1) and (15), see the critical discussion in Sec. 8 of Ref. [6].) as it should be. Finally, we want to stress that the overdampedlimit has to be precisely defined by rescaling Eq. (1) and (15), see the critical discussion in Sec. 8 of Ref. [6].