Brownian system in energy space

Eur. Phys. J. Special Topics 216, 37–48 (2013)© EDP Sciences, Springer-Verlag 2013DOI: 10.1140/epjst/e2013-01727-1

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Brownian system in energy space

Non-equilibrium distribution function in energy representation

B.I. Leva

Bogolyubov Institute for Theoretical Physics, National Academy of Science of Ukraine,Metrolohichna 14-b, Kyiv 03680, Ukraine

Received 30 November 2012 / Received in final form 6 December 2012Published online 31 January 2013

Abstract. The main goal of this article is to present a simple way todescribe non-equilibrium systems in energy space and to obtain newspacial solution that complements recent results of B.I. Lev and A.D.Kiselev, Phys. Rev. E 82, (2010) 031101. The novelty of this presen-tation is based on the kinetic equation which may be further usedto describe the non-equilibrium systems, as Brownian system in theenergy space. Starting with the basic kinetic equation and the Fokker-Plank equation for the distribution function of the macroscopic systemin the energy space, we obtain steady states and fluctuation relationsfor the non-equilibrium systems. We further analyze properties of thestationary steady states and describe several nonlinear models of suchsystems.

1 Introduction

According to basic principles of thermodynamics, when a macroscopic system isbrought into contact with a thermal bath, the system evolves in time approach-ing the equilibrium state in the course of relaxation. The state of equilibrium is welldefined only under certain idealized conditions [1–3], so that the properties of suchsystem are determined by its peculiarities and characteristics of the thermal bath.In most cases, however, the systems are subjected to non-equilibrium conditions andexternal constraints [4–8]. Therefore, it is difficult, if not impossible, to determine thegoverning parameters that can be held constant. Nevertheless, there exist stationarystates that can be unambiguously defined for certain open systems [4]. Examples ofsuch systems are given by hot electrons in semiconductors [9], a system of photons oninhomogeneous scatterers, when the diffraction coefficient depends on the frequencyof photons [10,11], a system of high-energy particles in accelerators that originatesfrom collision with macroscopic particles in dusty plasma [12].Currently, there does not exist a well-developed description method of the non-

equilibrium distribution function, which would take into account possible systemstates. A standard method describing non-equilibrium states is based on the infor-mation on the equilibrium state and small deviations from this state. Well-known

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38 The European Physical Journal Special Topics

fluctuation-dissipation theorem describes possible small fluctuations around the equi-librium state. The non-equilibrium in this approach is treated as a small modifica-tion of the equilibrium distribution function. Although far-from-equilibrium systemsare abundant in nature, there is no unified commonly accepted theoretical approachwhich determines possible states of such systems. Hence, it is a fundamentally impor-tant task to develop a method for exploring general properties of stationary statesof open systems and to establish conditions of their existence. Recent approacheswith experimental accessibility of the generalized fluctuation-dissipation relations forthe non-equilibrium steady states have been used as a possible description of a non-equilibrium system [13]. The solutions to this problem are the modified fluctuation-dissipation and Einstein relations at the non-equilibrium steady states [14–17] or anexactly solvable canonical model [18,19].Every system in equilibrium can be described in the phase space with the help of

the distribution function which depends on the states in this space. In equilibriumstatistical mechanics, these states are known to be generally described in terms ofenergy surfaces giving, for certain systems, micro-canonical and canonical ensembles[1–3]. There may be considered a natural hypothesis, that the non-equilibrium dis-tribution function will depend on the energy of the system. In a similar way, ourconsiderations will be based on the energy representation, where the states of thesystem are determined solely by their energies [20]. The variation of system energycan induce the variation of the state of the macroscopic system. In general case, thenon-equilibrium system can transit from one energy value to another one and thisprocess depends on the external effect and initial conditions. The external effect ispresent first of all in the variation of system energy, the energy also changes if thesystem dissipates or absorbs the energy as a result of the initial peculiarity of thebehavior. The main idea of this article consists in the description of the evolutionof a non-equilibrium system as a possible Brownian motion of the system betweendifferent states with dissipation energy and diffusion in the energy space. For brevity,such systems will be referred to as Brownian systems. In spite of the fact that theBrownian motion has long been the subject of intense studies (recent review on itshistory can be found in [21] or [22]) it is still interesting to understand the behavior ofthe Brownian system driven far from the equilibrium. It should be emphasized thatthe theory of the Brownian motion can be applied to the non-equilibrium systemstoo.Typically, the interaction between the Brownian system and the environment in-

volves the process of direct energy interchange. During this process, the dissipationmay take the energy away from the system leading to a loss of its energy. This is apositive friction. In the opposite case of negative friction, the transfer of energy fromthe thermostat results in the energy input. In addition to the deterministic part ofinteraction, there are fluctuation effects of the environment that affect the system,giving rise to a rapid change of its state. Such changes may take place when thestructure of the environment is complicated due to the presence of additional systemsor some of its characteristics can be directly exposed by the processes running in theBrownian system. So, the above pattern suggests the way of describing the behaviorof the Brownian systems interacting with surrounding media, using the stochasticdynamics. Therefore, there is an ensemble of such systems characterized by the prob-ability distribution function which, in particular, defines the stationary steady states,formed under non-equilibrium conditions. Typically, in the far from equilibrium sys-tems, the distribution functions of stationary states much differ from the well-knownequilibrium distributions. So, to describe the class of systems, called the Browniansystems, we suggest using the Langevin dynamics in the energy representation.The main goal of this article is to present a simple way to describe the non-

equilibrium systems in the energy space and to obtain a new special solution, which

From Brownian Motion to Self-Avoiding Walks and Levy Flights 39

was not described recently in [20]. Doing so, we will present general ideas about energyrepresentation of states of non-equilibrium systems. The novelty of this presentationis based on the kinetic equation which may be further used to describe the non-equilibrium systems, as Brownian system in the energy space. In this article, wehave extended and simplified the approach of Ref. [20] to the description of statesof the system when the dissipation and absorption processes as well as the effectof the environment are taken into account. Based on the basic kinetic equation andthe Fokker-Plank equation for the distribution function of a macroscopic system inthe energy space, we will obtain the steady states and the fluctuation relations forseveral non-equilibrium systems. The properties of the stationary steady states ofsuch systems will be our primary concern. A few nonlinear models of systems withdifferent processes will be described.

2 Fokker-Planck equation in energy presentation

It is well known that any state of a system can be described in terms of the distributionfunction, which determines all thermodynamic properties of a macroscopic system. Inpractice, the statistical description of a macroscopic system requires the knowledgeof only a few parameters, energy being one of them. In the case when there is noinformation on the system, it is possible to determine the system state terms of energy.The energy variation of the system defines the change of the states of the system.The energy representation, where the states of the system are distinguished solelyby their energies, can be viewed as a very basic description of the non-equilibriumsystem behavior, under conditions, when other dynamical variables, for some reasons,are irrelevant and can be disregarded. The non-equilibrium distribution function canbe defined as ρ(ε, t) which takes into account the dependence on a system energy εand on time. The energy distribution function, in the general case, can be obtainedfrom the basic kinetic equation, which presents the system evolution during a longtime and takes into account possible fast and slow processes. In terms of energy, thebasic kinetic equation for the non-equilibrium distribution function can be obtainedin a general form

∂ρ(ε, t)

∂t=

∫{W (ε, ε′)ρ(ε′, t)−W (ε′, ε)ρ(ε, t)} dε′, (1)

where W (ε, ε′) is the probability of the transition between different energies of a sys-tem. The basic kinetic equation represents the balance equation for the probabilityof possible states. The energy presentation of non-equilibrium processes is valid onlyif the considered variable is canonical and carries the average in the phase as theconjugate value. The basic kinetic equation leads to determination of a system evo-lution for a long time. All solutions of a basic kinetic equation for the infinite timehave fundamental properties: such solutions reduce to the stationary solutions for anysystem. This stationary solution corresponds to the law of entropy increase [24]. In aspecial case, for independent different states, if W (ε, ε′) is the operator of the secondrank, the basic kinetic equation becomes the Fokker-Planck equation

∂ρ(ε, t)

∂t=∂

∂εA(ε)ρ(ε, t) +

1

2

∂2

∂ε2D(ε)ρ(ε, t). (2)

The Fokker-Planck equation is a special case of the basic kinetic equation and is used,in generally, as an approximated form for it. The coefficients A(ε) and D(ε) in ourcase depend on the energy, and their physical sense should be determined for different


situations. The physical sense of these coefficients can be clarified if we return to thedynamic equation.The next step consists in the determination of the dynamic equation for a system

energy. This dynamic equation will be used in the form of the Langevin equation. Inthe non-equilibrium case, there is a dissipation of the energy which can be consideredwithin the energy variation law at an external effect. The energy evolution of thesystem can be described in terms of the dissipation energy and the random walkof a system in the energy space. System random walk is the result of interactionbetween the system and the environment. The environment effect should be includedto ensure the random behavior of changes in a system energy. The system cannotdeduce instantly after fast change of the environment media and should relax toa new state. This process takes into account possible degradation of the system,which is in contact with the environment. Such a system will be called the Browniansystem. From this position we can describe the system non-equilibrium fluctuationsand, probably, determine the system new state for this condition. The general dynamicequation, which describes the variation of the energy and takes into account the energydissipation and the random walk in the energy space, can be used in the form of theLangevin equation with the multiplicative noise:

∂ε

∂t= f(ε) + g(ε)L(t), (3)

where the second part takes into account the non-linearity of the diffusion process.General dynamic equation by form is the same as the usual Langevin equation inthe energy space, where f(ε) is the function determining the rate of direct energyexchange and g(ε) is the energy diffusion function; L(t) represents the Gaussian whitenoise. The Langevin equation by form is different from the Fokker-Planck equation,but both equations are equal in the mathematical sense. In this case, for the trueprobability distribution function we can write a different equation, according to thephysical process. In general case, there are two different approaches. If we considerthe dependence of the coefficient g(ε) only on the energy in the staring point, wecan obtain the equation for the non-equilibrium distribution function in the Ito formEq. (2), with the diffusion coefficient D(ε) = σ2g2(ε), where σ2 is intensity of whitenoise. If g(ε) depends on the energy before and after jump, we derive the diffusiveequation in the Stratonovich form:

∂ρ(ε, t)

∂t= − ∂∂ε(f(ε)ρ(ε, t)) +

σ2

2

∂

∂εg(ε)

∂

∂εg(ε)ρ(ε, t). (4)

Realization of different states for any system, in general case, is determined by theprevious state of the system and can form the possible future state. We will usethe Stratonovich presentation. However, we note that there exists a simple way oftransforming between these two different presentations [23,24]. The Fokker-Planckequation can be rewritten in a more usual form of the local conservation law:

∂ρ(ε, t)

∂t=∂

∂εJ(ρ(ε, t)). (5)

The probability flow may be presented as:

J = −(f(ε)− σ

2

2g(ε)

∂

∂εg(ε)

)ρ(ε, t) +

σ2

2g2(ε)

∂

∂ερ(ε, t). (6)

The stationary solution for J(ρ(ε, t)) = 0 is:

ρs(ε) = A exp

{∫ ε0

2f(ε′)dε′

σ2g2(ε′)− ln g(ε)

}. (7)


This stationary solution can be considered as the distribution function with regard tothe result of nonlinear properties of the environment and of the system. The steadystate of the system can be determined from the condition of the extremum of the dis-tribution function. This approach would enable us to select the states with a thermo-dynamically stable distribution function. The stationary solution in the generalnon-equilibrium case can be presented in the form:

ρs(ε) = A exp {−U(ε)} , (8)

where

U(ε) = ln g(ε)−∫ ε0

2f(ε′)dε′

σ2g2(ε′)· (9)

The extremal value of this distribution function can be obtained from the equation

U ′(ε) =1

D(ε)(D′(ε)− f(ε)) = 0, (10)

where the prime ′ signifies the derivative with respect to energy. This equation canbe regarded as the condition of diffusion-drift balance

D′(ε) = f(ε) (11)

that determines the relation between the dissipation in the system and the diffusionin the energy space for the stationary case. The shape of the distribution functionis determined by the effective energy potential U(ε). In particular, the distributionfunction reaches its extreme value at the energies determined by the stationary pointsof the potential U(ε). These points can be found from the stationary equation thatcan be regarded as the condition of the diffusion-drift balance, between the diffusionover states of the environment and the dissipation in the system. This balance con-dition gives the value of the most probable steady-state energy, which correspondsto the extremum of the energy potential U(ε). Of course, the obtained equation canhave more than one solution,but from physical viewpoint we can select the solutionfor which the distribution function has the greatest value. In the case where moresolutions are found, we should expand around every stationary point. In the case ofone solution, the stationary non-equilibrium distribution function can be presentedin the form

ρs(ε) = exp {−U(ε)} exp(−U ′′(ε)ε2) , (12)

where

−U ′′(ε) = 1

D(ε)(D′′(ε)− f ′(ε)) . (13)

In the vicinity of the most probable energy, the steady-state distribution can be ap-proximated by the Gaussian function where there is the second derivative of thepotential with respect to the energy. There is a variety of typical cases representingthe newly formed steady states of non-equilibrium systems and depending on theexchange and diffusion functions, f(ε) and g(ε). This stationary solution presentsthe Gaussian distribution function and rounds the new value of system energy, whichproduces the dissipation and diffusion in the energy space. If we assume that the dissi-pation of the system f(ε) is the nonlinear function of state, we can obtain many inter-esting situations, including the noise-induced transition in the new non-equilibriumstate and which will be more stable than the previous one. This state is caused by thespatial condition but a system can realize the new properties, which are not realizedfor the initial conditions. Herein below we discuss some of the most important cases.


3 The steady states and fluctuation relations

We begin with the noiseless case and an assumption on the singular limit of thevanishing diffusion, g(ε) = 0. Then, the temporal evolution of the energy distributionfunction, initially prepared at ε = ε0 with ρ0(ε, 0) = δ(ε− ε0), is as follows: ρ(ε, t) =δ(ε− ε(ε0, t)), where ε(ε0, t) is the solution of the initial value problem,

∂ε

∂t= −f(ε), ε(0) = ε0. (14)

Let us suppose that there is a local minimum of the potential located at ε = εs.Then, the energy εs is the attractive stationary equilibrium point that defines thestationary distribution ρs(ε) = δ(ε− εs, t)). This implies that, when the initial valueof the energy ε0, falls within the corresponding basin of attraction, the distributionfunctions evolve in time approaching the steady state: ρ(ε, t) = δ(ε−εs, t)) as t tendsto infinity.Note that the steady-state function takes the form of the micro-canonical distri-

bution, parameterized by the energy value ε0, only if the energy is conserved, anddiffusion is absent, g(ε) = 0. In the conservative system for the f(ε) = 0, the sta-tionary solution transforms to the constant. The equation for the non-equilibriumdistribution function in this case has the form of the diffusive equation, which hasthe known solution

ρ(ε) = A1√4πσ2c

exp

(− ε

2

4σ2t

)(15)

that describes the free migration of a system (fuzzy in the energy space). The measureof the fuzzy increases in the time according to the

⟨ε2⟩= 4σ2t. This solution presents

the system evolution which at the initial state is described by the equilibrium distrib-ution function ρ(ε) = δ(ε−ε0). All system states at the initial time are at the point ofthe surface of energy conservation. Fluctuations of the external media are manifestedin smearing of the micro-canonical distribution and have uniform distribution at allpossible energies. It represents the situation, when the stationary probability densityfunction does not exist. In this case we should take into account the fact that ε = 0 isnot only the inner limit but also the stationary point and the diffusion transmutes intozero. This point is natural, attractive boundary and the whole probability density hasan extremum indeed in this point. This follows from the normalization condition [15].In the case of energy conservation f(ε) = 0, the stationary solution can be presentedin the simple form:

ρs(ε) = A exp {−ln g(ε)} (16)

that corresponds to the canonical equilibrium distribution function only in the case ifg(ε) = eβε where β is the inverse temperature. The only possible diffusion coefficientdependence on the energy is exponential:

D(ε) = σ2g2(ε) = σ2e2βε. (17)

The exponential dependence may emerge as a special feature of interaction betweenthe system and the environment.We can also consider the system when the multiplicative noise is absent g(ε) = 1

and the Langevin equation can be written in a more simple form

∂ε

∂t= f(ε) + L(t), (18)


where the dissipation of the system f(ε) determines the energy change at differentprocesses, which can take place in the system, and L(t) determines the possible fluc-tuations which may not describe the introduced previous part and may not take intoaccount the random environment. The dependence of the dissipation function on theenergy describes all possible processes which may take place under the restrictionof the external condition. Below we will show different interesting cases when thisfunction depends on energy. It is important to note that in all cases this functionshould depend on the initial value of the energy, inasmuch as the system does notknow where the transformation is. Correlation between the two values of fluctua-tions L(t) at the two different moments of time is not zero only for the time intervalwhich is equal to the time of action. It is possible to describe it by the relation〈L(t1)L(t2)〉 = φ(t1 − t2). The function φδ(t1 − t2) should have a sharp peak andsatisfy the condition

∫φ(τ)dτ = σ2 for the white noise. To describe an impact of the

white noise, we will use the simple Fokker-Plank equation for the non-equilibriumdistribution function in the form [23,24]:

∂ρ(ε, t)

∂t= − ∂∂ε(f(ε)ρ(ε, t)) +

σ2

2

∂2ρ(ε, t)

∂ε2· (19)

This equation can be rewritten in the form of the local conservation law for thedistribution function:

∂ρ(ε, t)

∂t=∂J(ε)

∂ε, (20)

where the flux can be presented as:

J(ε) = −f(ε)ρ(ε, t) + σ2

2

∂ρ(ε, t)

∂ε· (21)

The stationary solution of this equation when J(ε) = 0 has a simple form:

ρ(ε) = A exp

(∫ ε0

f(ε′)σ2dε′)· (22)

In the case when the energy dissipation can be presented as the function f(ε) = −γε,which takes into account the possible dissipation of the system under the externalcondition, the stationary solution can be rewritten in the other form:

ρ(ε) = A exp

(−γε

2

σ2

)· (23)

To determine the physical meaning of the coefficient of the distribution we shouldreturn to the dynamic equation for the energy Eq. (18). The solution of this equationhas the form: ⟨

ε2⟩= ε20 exp(−2γt) +

σ2

2γ(1− exp(−2γt)) (24)

with the necessary condition limt→∞⟨ε2⟩= σ2/2γ. In this definition, we can rewrite

the non-equilibrium distribution function in the form:

ρ(ε) = A exp

(− ε

2

〈ε2〉), (25)

which takes the form of the equilibrium “Maxwell distribution function”. It is thesame as in the equilibrium case. Contrary to the noiseless case, at g(ε) = 0, we


can have the steady state even without equilibrium. An important example is thecanonical equilibrium Boltzmann-Gibbs distribution.We can note the generalization of the fluctuation dissipative theorem in the energy

representation. We need further consideration of the most general case, when thesystem is in contact with spatial external medium. In such a case, the average of thedispersion energy fluctuation can be written in the form:

⟨ε2⟩=σ2

2g2(0). (26)

In the equilibrium case, the ratio between energy and temperature fluctuations canbe written in the well-known form [3]

⟨ε2⟩=α(0)

β, (27)

where α(0) is the sensitivity of the system, which for energy fluctuations at theequilibrium case, presents the heat capacity cv of the system and β is the inversetemperature. In the non-equilibrium nonlinear case, we can introduce, as a postulate,the general presentation of the sensitivity of the system in the form:

α(ε) =βσ2

2g2(ε). (28)

It takes into account the possible dependence of a system reaction on the externaleffect. This consideration should take into account the physical condition and thepeculiarity of the interaction of the non-equilibrium system with the environmentand the degradation process in the system. Fluctuations of the environment determinethe system temperature and all possible states in this system. The temperature ofthe system is determined from the diffusion in the energy space and this diffusion is theuniversal characteristic of the environment.To conclude this part, we can make another important note. From the theory of

Markovian processes it is known that if we consider the fluctuations of the differentcoefficients in the function f(ε), as a result, we can obtain the diffusion coefficienttoo. As an example, we can examine the case when the function f(ε) takes the formf(ε) = αte

βε. The Langevin equation for the energy Eq. (3) in this case can be writtenas:

de−βε(t)

dt= −βαt , (29)

where αt = α + ξt has the constant part and the part ξt which describes the effectof the white noise of the environment [15]. If we introduce the new variable z =e−βε, the Fokker-Plank equation for the non-equilibrium distribution function will betransformed to

∂ρ(z, t)

∂t=∂

∂z(αβρ(z, t)) +

σ2β2

2

∂2

∂z2ρ(z, t) (30)

which has the stationary solution

ρs(s) = exp

(2α

σ2βz

)· (31)

In the case when βε > 1 we have

ρs(ε) = exp {−βε} (32)

when 2ασ2β= 1. This solution is similar to the standard solution.


4 Usual Brownian motion

The energy representation is also applicable in the case of the ordinary Brownianparticle description [20]. The dynamics of the Brownian particle is usually describedin terms of velocity v by the Langevin equation

∂v

∂t= −γv + F (t), (33)

where γ is the friction coefficient and F (t) is the random force which describes theaction on the particle from the fluid. The conditions for the averages with regardto the equilibrium ensemble are 〈F (t)〉 = 0 and 〈F (t)F (0)〉 = φ2δ(t). The last onesatisfies the condition of the white noise and describes the non-correlation processof the particle motion. For the free Brownian particle there exists only the kineticenergy ε = Mv2?2. From this representation of the energy of the Brownian particle,the energy evolution can be derived as follows:

∂ε

∂t=Mv

∂v

∂t= −2γε+

√2MεF (t). (34)

This is equivalent to general representation of Langevin dissipative equation withf(ε) = −2γε, g(ε) = √ε and L(t) = √2MF (t). Using the well-known solution of theLangevin equation for the velocity we can obtain

⟨v2⟩=φ2

2γ=kT

M

which can be transformed to the other well-known relation 〈ε〉 = kT/2, where theEinstein relation between the temperature and the friction constant is used. Fromthe solution of the Langevin equation in the energy presentation one can obtain thefollowing relations √

〈ε2〉 = φ2

4γ2M = kT

that satisfy the equilibrium condition. For the ordinary Brownian particle, we candetermine the stationary solution as

ρ(ε) = A exp

(−4γφ2ε−ln√ε

)=A√εexp (−βε) . (35)

The obtained stationary solution completely reproduces the well-known equilibriumdistribution function for an ordinary Brownian particle.The above considerations are also applicable to the systems of Brownian particles

in a random inhomogeneous environment. In such environment, some characteristicssuch as coupling constants and the friction coefficient may contain the stochastic in-duced contributions and thus become random variables. The examples include largeparticles in the inhomogeneous environment, impurity particles placed into the dustyplasma, as well as the systems whose kinetic properties non-linearly depend on a par-ticle velocity or energy. In conclusion, let us consider a very simple problem of motionof the Brownian particle in the heterogeneous medium. In this case, the characteristicof the medium can be taken into account by different values of the friction coefficient,which depend on the spatial position as the random quantity. We can use the generalresults of [15] where the present approach is used in describing the noise inducedphase transition. Then, the Langevin equation can be written in the form

∂v

∂t= −γv, (36)


where γt = γ + ξt consists of the constant part γ, which determines the average fric-tion coefficient and of the chaotic part ξt which describes the effect of the randomchange of the friction of a medium. In the case of the white noise of possible fluc-tuations of the density or other parameters, which characterize a medium, we canuse the Fokker-Plank equation for the non-equilibrium distribution function in theStratonovich interpretation in the standard form [15]:

∂ρ(v, t)

∂t=∂

∂v(γvρ(v, t)) +

σ2

2

∂2

∂v2v2ρ(v, t). (37)

The stationary solution of this equation has the form [15]:

ρs(v, t) = Nv−( 2γ

σ2+1). (38)

This result can be proved by a direct substitution of the solution in the previousequation. This stationary solution differs from the solution in the standard case, whenthe diffusion coefficient and friction coefficient do not depend on spatial position anddo not fluctuate. This result can be examined experimentally.One can consider the case when the standard Langevin equation also contains an

additional part of the thermal motion of the Brownian particle. Then, we can rewritethe Fokker-Plank equation in the other form:

∂ρ(v, t)

∂t=∂

∂v

(γ +σ2

2

)vρ(v, t) +

∂2

∂v2

(γ

βM+σ2

2v2)ρ(v, t). (39)

The stationary solution in this case can be obtained in the form

ρs(v, ) = N exp−⎧⎨⎩∫ v0

(γ + σ

2

2

)vdv

γβM+ σ

2

2 v2

⎫⎬⎭ · (40)

This solution can be presented in the other form:

ρs(v, ) = N

(σ2

2γβM

) 12

exp−{(2γ

σ2+ 1

)ln

(1 +σ2

2γβMv2

)}· (41)

In the case σ2

2γ � 1, we can present

ln

(1 +σ2

2γβMv2

)=σ2

2γβMv2

and reproduce the standard Maxwell distribution function

ρs(v, t) = N exp

(−Mv

2

kT

)·

In the opposite case, the other distribution function is recovered. This new solutioncan be presented in the form:

ρs(v) = N

(σ2

2γβM

) 12(1 +σ2

2γβMv2

)− 12 ( 2γσ2+1)(42)

which in the case σ2/2γ � 1, reproduces the constant distribution function. Thistheoretical prediction can be examined experimentally.


5 Conclusion

In this article, a general description of the non-equilibrium distribution function inthe energy space has been proposed. Based on the Fokker-Plank equation for thedistribution function of the macroscopic system, the steady states and fluctuationrelations were obtained. The proposed approach takes into account the possible mo-tion of the system between different states by dissipation and diffusion in the energyspace. Thus, there has been proposed a model describing the non-equilibrium sys-tems with the purpose of determining the new stationary states which are far fromequilibrium. We obtained the stationary distribution functions of non-equilibriumsystems in contact with a nonlinear environment for various mechanisms of energyabsorption and dissipation. The nonlinear models which describe a possible station-ary non-equilibrium state have been described and possible experimental results werepredicted. A similar approach was used to describe the behavior of the dust parti-cles in weakly ionized plasma [12,26,27] and it was shown that the particles havea diffirent effective temperature. These particles also form a periodical structure fordifferent thermodynamic parameters as in the case of a standard approach. Similarsimple cases can be experimentally verified as well.

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Brownian system in energy space