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Annals of Physics 316 (2005) 393-413 www.elsevier.com/locate/aop

Stationary solutions of Liouville equations for non-Hamiltonian systems
Vasily E. Tarasov
*
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia Received 14 August 2004; accepted 2 November 2004

Abstract We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems. ã 2004 Elsevier Inc. All rights reserved.
PACS: 05.20.-y; 05.20.Gg Keywords: Liouville equation; Non-Hamiltonian systems; Canonical distribution

1. Introduction The canonical distribution for the Hamiltonian systems was defined by Gibbs in the book ``Elementary principles in statistical mechanics'' [1], published in 1902. In general, classical systems are not Hamiltonian systems and the forces are the sum of
*

Fax: +7095 9390397. E-mail address: tarasov@theory.sinp.msu.ru.

0003-4916/$ - see front matter ã 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.11.001


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potential and non-potential forces. Non-Hamiltonian and dissipative systems can have the same distributions as Hamiltonian systems. The canonical distributions for the non-Hamiltonian and dissipative systems were considered in [2-5,7,8,6,9-11]. The aim of this work is the extension of the statistical mechanics of conservative Hamiltonian systems to a wide class of non-Hamiltonian and dissipative systems. Let us point out non-Hamiltonian systems with distribution functions that are defined by the Hamiltonian. (1) In the papers [2-6,9], the constant temperature systems with minimal Gaussian constraint are considered. These systems are the non-Hamiltonian systems that ?nî are described by the non-potential forces in the form Fi Ì Ðcpi and the Gaussian non-holonomic constraint. Note that this constraint can be represented as an addition term to the non-potential force. (2) In the papers [12,13], the canonical-dissipative systems are considered. These systems are the non-Hamiltonian systems that are described by the non-potential ?n î forces Fi Ì ÐoG?H î=opi , where G (H) is a function of Hamiltonian H. Note that the distribution functions are derived as solutions the Fokker-Planck equation. It is known that Fokker-Planck equation can be derived from the Liouville equation [14]. (3) In the paper [15], the systems with non-holonomic constraint and non-potential ?n î forces Fi Ì 0 are considered. The equations of motion for this system are incorrect [16]. The correct form of the equations is derived in [15] by the limit s fi 0. This procedure removes the incorrect term of the equations. (4) In the paper [11], the canonical distribution is considered as a stationary solution of the Liouville equation for a wide class of non-Hamiltonian system. This class is defined by a very simple condition for the non-potential forces: the power of the non-potential forces must be directly proportional to the velocity of the Gibbs phase (elementary phase volume) change. This condition defines the general constant temperature systems. Note that the condition is a non-holonomic constraint. This constraint leads to the canonical distribution as a stationary solution of the Liouville equations. For the linear friction, we derived the constant temperature systems. The general form of the non-potential forces is not derived in [11]. (5) In the paper [17], the quantum non-Hamiltonian systems with pure stationary states are considered. The correspondent classical systems are not discussed. (6) In the paper [19], the non-Gaussian distributions are suggested for the non-Hamiltonian systems in the fractional phase space. Note that non-dissipative systems with the usual phase space are dissipative systems in the fractional phase space [19]. Khintchin [20] revealed the deep relation between the Gaussian central limit theorem and canonical Gibbs distribution. However, the Gaussian central limit theorem is non-unique. Levy and Khintchin have generalized the Gaussian central limit theorem to the case of summation of independent, identically distributed random variables which are described by long tailed distributions. In this case, non-Gaussian distributions replace the Gaussian in the generalized limit theorems. It is interesting


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to find statistical mechanics and thermodynamics that is based on non-Gaussian and non-canonical distributions [21-24]. The aim of this paper is the description of non-Hamiltonian and dissipative systems with (canonical and non-canonical) distributions that are defined by Hamiltonian. This class can be described by the non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change must be directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by the Hamiltonian. These distributions can be derived analytically as solutions of the Liouville equation for non-Hamiltonian systems. The special constraint allows us to derive solutions for the system, even in far-from equilibrium states. This class of the non-Hamiltonian systems is characterized by the distribution functions that are determined by the Hamiltonian. The constant temperature systems [2-6,9], the canonical-dissipative systems [12,13], and the Fermi-Bose classical systems [13] are the special cases of suggested class of non-Hamiltonian systems. In Section 2, the definitions of the non-Hamiltonian and dissipative systems, mathematical background and notations are considered. In Section 3, we consider the condition for the non-potential forces. We formulate the proposition that allows us to answer the following question: Is this system a canonical non-Hamiltonian system? We derive the solution of the N-particle Liouville equation for the non-Hamiltonian systems with non-holonomic constraint. In Section 4, we consider the non-holonomic constraint for non-Hamiltonian systems. We formulate the proposition which allows us to derive the canonical non-Hamiltonian systems from the equations of non-Hamiltonian system motion. The non-Hamiltonian systems with the simple Hamiltonian and the simple non-potential forces are considered. In Section 5, we derive the class of non-Hamiltonian systems with canonical Gibbs distribution as a solution of the Liouville equation. In Section 6, we consider the non-Gaussian distributions as solutions of the Liouville equations for the non-Hamiltonian systems. In Section 7, we derive the analog of thermodynamics laws for the non-Hamiltonian systems with the distributions that are defined by Hamiltonian. Finally, a short conclusion is given in Section 8.

2. Definitions of non-Hamiltonian, dissipative, and canonical non-Hamiltonian systems Let us consider the definitions of non-Hamiltonian and dissipative classical systems [25], which are used for the formulation of our results. Usually a classical system is called a Hamiltonian system if the equations of motion are determined by Hamiltonian. The more consistent definition of the non-Hamiltonian system is connected with Helmholtz condition for the equation of motion. Definition 1. A classical system which is defined by the equations dqi Ì Gi ; dt dpi Ì F i; dt ? 1î


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where i = 1, ..., N, is called Hamiltonian system if the right-hand sides of Eq. (1) satisfy the Helmholtz conditions for the phase space oGi oGj Ð Ì 0; op j op i oF i oGj ? Ì 0; op j oqi oF i oF j Ð Ì 0: oqj oqi ? 2î

Here Gi = Gi (q, p), Fi = Fi (q, p, a, t), where a is a set of external parameters. If the Helmholtz conditions are satisfied, then the equations of motion for the system (1) can be represented as canonical equations dqi oH Ì ; op i dt dpi oH ÌÐ ; oqi dt ? 3î

which are completely characterized by the Hamiltonian H = H (q, p, a). In this case, the forces, which act on the particles are potential forces. If the functions Gi for the non-Hamiltonian system (1) are determined by the Hamiltonian Gi Ì oH op i ? 4î

and the Hamiltonian is a smooth function on the momentum space, then the first condition (2) is satisfied o2 H o2 H Ð Ì 0: op i op j op j op i In this case, the second condition (2) has the form oF i o2 H ? Ì 0: op j oqi op j ? 5î

In general, the second term does not vanish. For example, in the non-linear one-dimensional sigma-model [27] the second term of the left-hand side of Eq. (5) is defined by the metric. Definition 2. A mechanical system is called non-Hamiltonian if at least one of conditions (2) is not satisfied. Let us consider the time evolution of the classical state which is defined by the distribution function qN (q, p, a, t). The N-particle distribution function in the Hamilton picture (for the Euler variables) is normalized by the condition Z qN ?q; p; a; tîdN q dN p Ì 1: ? 6î The evolution equation of the distribution function qN (q, p, a, t) is Liouville equation in the Hamilton picture dqN ?q; p; a; tî Ì ÐX?q; p; a; tî qN ?q; p; a; tî: dt ? 7î

This equation describes the change of the distribution function qN along the trajectory in the 6N-dimensional phase space. Here, X is defined by


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X?q; p; a; tî Ì

oF i oG i ? : op i oqi

? 8î

Here and later we mean the sum on the repeated index i from 1 to N. Derivative d/dt is a total time derivative d o o o Ì ? Gi ? Fi : dt o t oqi op i If the vector function Gi is defined by Eq. (4), then X?q; p; a; tî Ì oF i o2 H ? : op i oqi op i ? 9î

In general, the second term does not vanish, for example, in the non-linear one-dimensional sigma-model [27]. In the Liouville picture (for the Lagrange variables) the function X defines the velocity of the phase volume change [26] Z dV ph ?a; tî Ì X?q; p; a; tîdN q dN p: dt Definition 3. If X 6 0 for all phase space points (q, p) and X < 0 for some points of phase space, then the system is called a dissipative system. We can define dissipative system using a phase density of entropy S ?q; p; a; tî Ì Ðk ln qN ?q; p; a; tî: This function usually called the Gibbs phase. Eq. (7) leads to the equation for the entropy density (Gibbs phase) dS ?q; p; a; tî Ì k X?q; p; a; tî: dt ?10î

It is easy to see that the function X is proportional to the velocity of the phase entropy density change. Therefore, the dissipative systems can be defined by the following equivalent definition. Definition 4. A system is called a generalized dissipative system if the velocity of the entropy density change does not equal to zero. Let us define the special class of the non-Hamiltonian systems with distribution functions that are completely characterized by the Hamiltonian. These distributions can be derived analytically as stationary solutions of the Liouville equation for the non-Hamiltonian system. Definition 5. A non-Hamiltonian system will be called a canonical non-Hamiltonian system if the distribution function is determined by the Hamiltonian, i.e., qN (q, p, a) can be written in the form qN ?q; p; aî Ì qN ?H ?q; p; aî; aî; ?11î


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where a is a set of external parameters. Examples of the canonical non-Hamiltonian systems: (1) The constant temperature systems [2-6,9] that have the canonical distribution. In general, these systems can be defined by the non-holonomic constraint, which is suggested in [11]. (2) The Fermi-Bose canonical-dissipative systems [13] which are defined by the distribution functions in the form qN ?H ?q; p; aîî Ì 1 : exp?b?H ?q; p; aîÐ lî? sî ?12î

(3) The classical system with the Breit-Wigner distribution function is defined by qN ?H î Ì k ?H Ð Eî ??C=2î
2 2

:

?13î

3. Distribution as a solution of the Liouville equation 3.1. Formulation of the results Let us formulate the proposition that allows us to answer the following question: Is this system a canonical non-Hamiltonian system? Let us consider the N-particle non-Hamiltonian systems which are defined by the equations dri oH Ì ; opi dt dpi oH ?nî ÌÐ ? Fi : ori dt oH : opi ?14î

The power of non-potential forces is defined by P?r; p; aî Ì Fi
?n î

?15î

If the power of the non-potential forces is equal to zero ?P Ì 0î and oH/ot = 0, then classical system is called a conservative system. The velocity of an elementary phase volume change X is defined by the equation X?r; p; aî Ì oFi o2 H oF ? Ì i: opi or i opi opi
?nî

?16î

We use the following notations for the scalar product: N oAi X oAxi oAyi oAzi Ì ? ? : oai oaxi oayi oazi i Ì1 The aim of this section is to prove the following result.


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Proposition 1. If the non-potential forces Fi satisfy the constraint condition g?H îFi
?nî

?nî

of the non-Hamiltonian system (14)

oH oFi Ð Ì 0; opi opi

?n î

?17î

then this system is a canonical non-Hamiltonian system with the distribution function qN ?r; p; aî Ì Z ?aî exp?ÐL?H ?r; p; aîîî; where the function L (H) is defined by the equation g ?H î Ì o L? H î : oH ?19î ?18î

The condition (17) can be formulated in other words: If velocity of the elementary phase volume change X is directly proportional to the power P of non-potential forces ?nî Fi of the non-Hamiltonian system (14) and coefficient of this proportionality is a function g (H) of Hamiltonian H, i.e., X?r; p; tîÐ g?H îP?r; p; tî Ì 0; then this system is a canonical non-Hamiltonian system. Note that any non-Hamiltonian system with the non-holonomic constraint (20) or (17) is a canonical non-Hamiltonian system. Example. Let us consider g (H)= 3Nb (a), where b (a)= 1/kT (a). This case is considered in [11]. If we consider the N-particle system with the Hamiltonian H ? r ; p; aî Ì
N X p2 i ? U ? r ; aî 2m iÌ1

?20î

?21î

and a linear friction, which is defined by the non-potential forces Fi Ì Ðcpi ; then the non-holonomic constraint (17) has the form
N X p2 i Ì kT ?aî; m iÌ1 ?nî

?22î

?23î

i.e., the kinetic energy of the system must be a constant. The constraint (23) is a nonholonomic minimal Gaussian constraint [11,9]. If the function g (H) is defined by g (H) = 3Nb (a), then the non-Hamiltonian system can have the canonical Gibbs distribution [11]. The classical systems that are defined by Eqs. (21)-(23) are canonical non-Hamiltonian systems.


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3.2. Proof of the result Solving the Liouville equation with the non-holonomic constraint (17), we can obtain the (canonical and non-canonical) distributions that are defined by the Hamiltonian. Let us consider the Liouville equation for the N-particle distribution function qN = qN(r, p, a, t). This distribution function qN express a probability that a phase space point (r, p) will appear. The Liouville equation for this non-Hamiltonian system oqN o o ? ?Gi qN î? ?Fi qN î Ì 0 or i opi ot expresses the conservation of probability in the phase space. Here, we use Gi Ì oH ; opi Fi Ì Ð oH ?n î ? Fi : or i ?24î

We define a total time derivative along the phase space trajectory by d o o o Ì ? Gi ? Fi : dt ot or i opi Therefore Eq. (24) can be written in the form (7) dqN Ì ÐXqN ; dt ?26î ?25î

where the omega function is defined by Eq. (16). In classical mechanics of Hamiltonian systems the right-hand side of the Liouville equation (26) is zero, and the distribution function does not change in time. For the non-Hamiltonian systems (14), the omega function (16) does not vanish. For this system, the omega function is defined by Eq. (16). For the canonical non-Hamiltonian systems, this function is defined by the constraint (17) in the form ?n î o H X Ì g?H îFi : op i In this case, the Liouville equation has the form dqN Ì Ðg?H îF dt
?nî i

oH q: opi N

?27î

Let us consider the total time derivative of the Hamiltonian. Using equations of motion (14), we have dH oH oH oH oH oH ?nî oH ?nî oH Ì ? ? Fi ?Ð ? Fi Ì : dt ot op i or i ori opi ot opi If oH/ot = 0, then the power P of non-potential forces is equal to the total time derivative of the Hamiltonian F
?nî i

oH dH : Ì opi dt


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Eq. (27) can be written in the form dqN dH q: Ì Ðg?H î dt N dt Let us consider the following form of this equation: dln qN dH : Ì Ð g ?H î dt dt o L? H î : oH ?29î ?28î

If g (H) is an integrable function, then this function can be represented as a derivative g ?H î Ì ?30î

In this case, we can write Eq. (29) in the form dln qN dL?H î : ÌÐ dt dt As a result, we have the following solution of the Liouville equation: qN ?r; p; aî Ì Z ?aî exp?ÐL?H ?r; p; aîîî: ?32î ?31î

The function Z (a) is defined by the normalization condition. It is easy to see that the distribution function of the non-Hamiltonian system is determined by the Hamiltonian. Therefore, this system is a canonical non-Hamiltonian system. Note that N is an arbitrary natural number since we do not use the condition N ) 1 or N fi 1.

4. Non-holonomic constraint for non-Hamiltonian systems 4.1. Formulation of the result Let us formulate the proposition which allows us to derive the canonical nonHamiltonian systems from any equations of motion of non-Hamiltonian systems. The aim of this section is to prove the following result. Proposition 2. For any non-Hamiltonian system which is defined by the equation dri oH Ì ; opi dt dpi Ì Fi ; dt ?33î

where Fi is the sum of potential and non-potential forces Fi Ì Ð oH ?n î ? Fi ; or i dpi ÌF dt ?34î

there exists a canonical non-Hamiltonian system that is defined by the equations dri oH Ì ; opi dt
new i

;

?35î


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where the non-potential forces Fnew are defined by i F
new i

Ì

Ak Ak dij Ð Ai Aj Ai Bj oH Fj Ð : Ak Ak Ak Ak opj
?n î

?36î

The vectors Ai and Bi are defined by the equations Ai Ì and Bi Ì og ? H î oH oH F oH or i opj
?nî j

og?H î oH oH F oH opi opj

?nî j

? g ?H î

oFj oH ? g ?H îF opi opj
?nî

?nî j

o2 Fj o2 H Ð opi opj opi op
?nî

?n î

?37î
j

? g ?H î

oFj oH ? g ?H îF or i opj

?nî j

o2 Fj o2 H Ð : ori opj ori opj

?38î

Note that the forces that are defined by Eqs. (36)-(38) satisfy the non-holonomic constraint (20), i.e., g?H îFnew j
new oH oF j o2 H Ð Ð Ì 0: opj or j opj opj

?39î

4.2. Proof. Part I In this section, we prove Eq. (36). Let us consider the N-particle classical system in the Hamilton picture. Denote the position of the ith particle by ri and its momentum by pi. Suppose that the system is subjected to a non-holonomic (non-integrable) constraint in the form f ?r; pî Ì 0: Differentiation of Eq. (40) with respect to time gives a relation Ai ?r; pî where Ai ?r; pî Ì of ; opi Bi ?r; pî Ì of : or i ?42î dpi dri ? Bi ?r; pî Ì 0; dt dt ?41î ?40î

An unconstrained motion of the ith particle, where i = 1, ..., N, is described by the equations dri Ì Gi ; dt dpi Ì Fi ; dt ?43î

where Fi is a resulting force, which acts on the ith particle. The unconstrained motion gives a trajectory which leaves the constraint hypersurface (40). The constraint forces Ri must be added to the equation of motion to prevent the deviation from the constraint hypersurface


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dri dpi Ì Gi ; Ì Fi ? Ri : ?44î dt dt The constraint force Ri for the non-holonomic constraint is proportional to the Ai [16] Ri Ì kAi ; ?45î where the coefficient k of the constraint force term is an undetermined Lagrangian multiplier. For the non-holonomic constraint (40), the equations of motion (43) are modified as dri dpi Ì Gi ; Ì Fi ? kAi : ?46î dt dt The Lagrangian coefficient k is determined by Eq. (41). Substituting Eq. (44) into Eq. (41), we get Ai ?Fi ? kAi î? Bi Gi Ì 0: Therefore, the Lagrange multiplier k is equal to kÌÐ Ai Fi ? Bi Gi : Ak Ak dpi Aj Fj ? B j Gj Ì Fi Ð Ai : dt Ak Ak dpi Ì Fnew i dt ?48î ?47î

As a result, we obtain the following equations: dri Ì Gi ; dt dri Ì Gi ; dt ?49î

These equations we can rewrite in the form (43) ?50î

with the new forces Ak Ak dij Ð Ai Aj Ai Bj Fj Ð Gj : Fnew Ì i Ak Ak Ak Ak

?51î

In general, the forces Fnew are non-potentials forces (see examples in [11]). i Eq. (49) are equations of the holonomic non-Hamiltonian system. For any trajectory of the system in the phase space, we have f = const. If initial values rk (0) and pk (0) satisfy the constraint condition f (r(0), p(0)) = 0, then solution of Eqs. (49) and (51) is a motion of the non-holonomic system. 4.3. Proof. Part II In this section, we prove Eqs. (37) and (38). Let us consider the non-Hamiltonian system (43) with Gi Ì oH ; opi Fi Ì Ð oH ?F or i
?nî i

?52î

and the special form of the non-holonomic constraint (40). Let us assume the following constraint: the velocity of the elementary phase volume change X(r,p,a) is directly proportional to the power P?r; p; aî of the non-potential forces, i.e.,


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X?r; p; aî Ì g?H îP?r; p; aî;

?53î

where g (H) depends on the Hamiltonian H. Therefore, the system is subjected to a non-holonomic (non-integrable) constraint (40) in the form f ?r; p; aî Ì g?H îP?r; p; aîÐ X?r; p; aî Ì 0: ?54î

This constraint is a generalization of the condition which is suggested in [11]. The ?n î power P of the non-potential forces Fi is defined by Eq. (15). The function X is defined by Eq. (16). Eq. (54) for the non-potential forces has the form g?H îFj
?n î

oH oF j Ð Ì 0: opj opj

?nî

Let us find the functions Ai and Bi for this constraint. Differentiation of the function f (r, p, a) with respect to pi gives ! ?nî of o o oFj ?nî oH Ai Ì Ì g?H îFj : Ð opi opi opj opi opj Therefore we obviously have (37). Differentiation of the function f (r, p, a) with respect to ri gives ! ?n î of o o oFj ?nî oH Bi Ì Ì g?H îFj : Ð or i or i opj or i opj Therefore, we have (38). 4.4. Minimal constraint models To realize simulation of the classical systems with canonical and non-canonical distributions, we must have the simple constraints. Let us consider the minimal constraint models which are defined by the simplest form of the Hamiltonian and the non-potential forces p2 ?n î H ?r; p; aî Ì ? U ?r; aî; Fi Ì Ðcpi : ?55î 2m P where p2 Ì N 1 p2 . For these models, the non-holonomic constraint is defined by the iÌ i equation f Ì g ?H î p2 Ð 3N Ì 0; m ?56î

where N is the number of particles. The phase space gradients (37) and (38) of the constraint can be represented in the form og ? H î p2 2pi og ? H î oH Ai Ì ? g ?H î ; Bi Ì : oH 2m oH or i m


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The non-potential forces of the minimal constraint models have the form Fi Ì Ð p2 dij Ð pi pj oU pi pj o g ? H î oU ?22 : 2 orj 2p ??p =2mî?og?H î=oH î? g?H îî oH orj p

It is easy to see that all minimal constraint models have the potential forces. For the minimal Gaussian constraint model og ? H î Ì 0; oH we have the non-potential forces in the form Fi Ì Ð oU p2 dij Ð pi pj : or j p2

This model describes the constant temperature systems [2-6,9,11]. 4.5. Minimal Gaussian constraint model Let us consider the N-particle system with the Hamiltonian H ? r ; p; aî Ì p2 ? U ? r ; aî ; 2m ?57î

the function g (H)= 3N/kT, and the linear friction Fi Ì Ðcpi ;
?nî

?58î

where i =1, ..., N. Note that N is an arbitrary natural number. Substituting Eq. (58) into Eqs. (15) and (16), we get the power P and the omega function X: PÌÐ c2 p; m X Ì Ð 3c N :

The non-holonomic constraint has the form p2 Ì kT ?aî; m i.e., the kinetic energy the friction parameter For the N-particle we have the following dri pi Ì; dt m ?59î of the system must be a constant. Note that Eq. (59) has not c. system with friction (58) and non-holonomic constraint (59), equations of motion: ?60î

dpi oU of ÌÐ Ð c pi ? k ; ori opi dt f ? r ; pî Ì 0:

where the function f is defined by f ?r; pî Ì 1?p2 Ð mkT î : 2 ?61î

Eq. (60) and condition (61) define 6N + 1 variables (r, p, k).


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Let us find the Lagrange multiplier k. Substituting Eq. (61) into Eq. (60), we get dpi oU ÌÐ ??k Ð cîpi : ori dt Using df/dt = 0 in the form p
i

?62î

d pi Ì0 dt 1 oU pj ? c: mkT orj d pi 1 oU o U pi pj Ì Ð : mkT orj ori dt

?63î

and substituting Eq. (62) into Eq. (63), we get the Lagrange multiplier k in the form kÌ

As a result, we have the holonomic system that is defined by the equations dri pi Ì; dt m ?64î

This system is equivalent to the non-holonomic system (60). For the classical N-particle system (64), condition (59) is satisfied. If the time evolution of the N-particle system is defined by Eq. (64) or Eqs. (60) and (61), then we have the canonical distribution function in the form q?r; p; a; T î Ì exp 1 ?F?a; T îÐ H ?r; p; aîî: kT ?65î

For example, the N-particle system with the forces Fi Ì x2 ?aî p p rj Ð mx2 ?aîr kT i j mx2 ?aîr2 : 2
i

?66î

can have canonical distribution (65) of the linear harmonic oscillator with U ?r; aî Ì

5. Canonical distributions In this section, we consider the subclass of the canonical non-Hamiltonian system that is described by canonical distribution. This subclass of the canonical non-Hamiltonian N-particle system is defined by the simple function g (H) = 3Nb (a) in the non-holonomic constraint (20). Proposition 3. If velocity of the elementary phase volume change is directly proportional to the power of non-potential forces, then we have the usual canonical Gibbs distribution as a solution of the Liouville equation. In other words, the non-Hamiltonian system with the non-holonomic constraint X Ì b?aîP ?67î


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can have the canonical Gibbs distribution qN Ì exp b?aî?F?aîÐ H ?r; p; aîî as a solution of the Liouville equation. Here, the coefficient b (a) does not depend on (r, p, t), i.e., db?aî=dt Ì 0: For the non-Hamiltonian systems, the omega function (16) does not vanish. Using Eq. (15), we have ?nî oH : ?68î X Ì b?aîFi opi In this case, the Liouville equation has the form dqN ?n î o H Ì Ðb?aîFi q: opi N dt Let us consider the total time derivative for the Hamiltonian dH oH Ì ?F dt ot
?nî i

?69î

oH : opi

?70î

If oH/ot = 0, then the energy change is equal to the power P of the non-potential ?nî forces Fi . Eq. (69) can be written in the form dqN dH q: Ì Ðb?aî dt N dt Therefore, the Liouville equation can be rewritten in the form dln qN ?r; p; a; tî dH ?r; p; aî ? b?aî Ì 0: dt dt Since coefficient b (a) is a constant (db (a)/dt = 0), we have d ?ln qN ?r; p; a; tî? b?aîH ?r; p; aîî Ì 0; dt i.e., the value (ln qN + bH) is a constant along the trajectory of the system in 6N-dimensional phase space. Let us denote this constant value by b?aîF?aî. Then, we have ln qN ?r; p; a; tî? b?aîH ?r; p; aî Ì b?aîF?aî; where dF?aî=dt Ì 0. It follows that: ln qN ?r; p; a; tî Ì b?aî?F?aîÐ H ?r; p; aîî: As a result, we have a canonical distribution function qN ?r; p; a; tî Ì exp b?aî?F?aîÐ H ?r; p; aîî in the Hamilton picture. The value F?aî is defined by the normalization condition (40). Therefore the distribution of this non-Hamiltonian system is a canonical distribution. ?71î


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Note that N is an arbitrary natural number since we do not use the condition N ) 1 or N fi 1.

6. Non-canonical distributions The well-known non-Gaussian distribution is the Breit-Wigner distribution. This distribution has a probability density function in the form q?xî Ì 1 : p?1 ? x2 î ?72î

The Breit-Wigner distribution is also known in statistics as Cauchy distribution. The Breit-Wigner distribution is a generalized form originally introduced [28] to describe the cross-section of resonant nuclear scattering in the form q?H î Ì k ?H Ð Eî ??C=2î
2 2

:

?73î

This distribution can be derived from the transition probability of a resonant state with known lifetime [29-31]. If the function g (H) of the non-holonomic constrain is defined by g?H î Ì 2? H Ð E î ?H Ð Eî ??C=2î
2 2

;

?74î

then we have non-Hamiltonian systems with the Breit-Wigner distribution as a solution of the Liouville equation. If the function g (H) of the non-holonomic constrain has the form g?H î Ì b?aî ; 1 ? a exp b?aîH ?75î

then we have classical non-Hamiltonian systems with Fermi-Bose distribution (12) considered by Ebeling [13]. This distribution can be derived as a solution of the Liouville equation. Note that Ebeling derives the Fermi-Bose distribution function as a solution of the Fokker-Planck equation. It is known that Fokker-Planck equation can be derived from the Liouville equation [14]. ?nî If the non-potential forces Fi are determined by the Hamiltonian F
?nî i

Ì ÐoG?H î=opi ;

?76î

then we have the canonical non-Hamiltonian systems, which are considered in [12,13]. These systems are called canonical dissipative systems. Note that the linear function g (H) in the form g?H î Ì b1 ?aî? b2 ?aîH leads to the following non-canonical distribution function: qN Ì Z ?aî exp Ð?b1 ?aîH ? 1b2 ?aîH 2 î: 2 ?77î


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409

The proof of this proposition can be directly derived from Eqs. (32) and (30). Let us assume that Eq. (11) can be solved in the form H Ì h?aîh?qN î; ?78î where h depends on the distribution qN. The function h (a) is a function of the parameters a. In this case, the function g (H) is a composite function R?qN î Ì Ðg?h?aîh?qN îî: This function can be defined by Ð1 1 oh?qN î h? aî : R?qN î Ì qN oqN dqN Ì R?qN îP: dt ?79î

?80î

In this case, the Liouville equation for the non-Hamiltonian system has the form ?81î

This equation is a non-linear equation. Note that the classical Fermi-Bose systems [13] have the function in the form R?qN î Ì Ðb?aî?qN Ð sq2 î: N ?82î

The non-linearity of the Liouville equation is not connected with an incorrectly defined phase space. This non-linearity is a symptom of the use of an incorrectly defined boundary condition. The Bogoliubov principle of correlation weakening cannot be used for classical Fermi-Bose systems. The classical Fermi-Bose systems can be considered as a model of open (non-Hamiltonian) system with the special correlation. Note that the non-linear evolution of statistical systems is considered in [33-39].

7. Thermodynamics laws for non-Hamiltonian systems Let us define the mean value f (a) of the function f (r, p, a) by the relation Z f ?aî Ì f ?r; p; aîqN ?r; p; aîdN r dN p and the variation for this function by n X of ? r ; p; aî dak : da f ? r ; p; aî Ì o ak k Ì1

?83î

?84î

The first law of thermodynamic states that the internal energy U (a) may change because of heat transfer dQ, and work of thermodynamic forces n X F k ? a î da k : ?85î dA Ì
k Ì1

The external parameters a ={a1,a2, ..., an} here act as generalized coordinates. In the usual equilibrium thermodynamics the work done does not entirely account for the


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change in the internal energy. The internal energy also changes because of the transfer of heat, and so dU Ì d Q Ð d A : Since thermodynamic forces Fk (a) are non-potential forces oF k ?aî oF l ?aî Ì ; oal oak ?87î ?86î

the amount of work dA depends on the path of transition from one state in parameters space to another. For this reason dA and dQ, taken separately, are not total differentials. Let us give a statistical definition of thermodynamic forces for the non-Hamiltonian systems in the mathematical expression of the analog of the first thermodynamics law for the mean values. It would be natural to define the internal energy as the mean value of Hamiltonian Z U ?aî Ì H ?r; p; aîqN ?r; p; aî dN r dN p: ?88î It follows that the expression for the total differential has the form: Z Z dU ?aî Ì da H ?r; p; aîqN ?r; p; aî dN r dN p ? H ?r; p; aîda qN ?r; p; aî dN r dN p: Therefore dU ?aî Ì oH ? r ; p; aî dak qN ?r; p; aî dN r dN p o ak Z ? H ?r; p; aîda qN ?r; p; aî dN r dN p: Z

?89î

In the first term on the right-hand side, we can use the definition of phase density of the thermodynamic force F ph ?r; p; aî Ì Ð k oH ?r; p; aî : o ak

The thermodynamic force Fk (a) is a mean value of the phase density of the thermodynamic force Z F k ?aî Ì F ph ?r; p; aîqN ?r; p; aî dN r dN p: ?90î k Using this equation we can pr Analyzing these expressions (89) answers for the work (85) the heat transfer is given by Z dQ Ì H ?r; p; aîda qN ?r; ove the relation (87). we see that the first term on the right-hand side of Eq. of thermodynamic forces (90), whereas the amount of p; aî d N r d N p: ?91î


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411

We see that the heat transfer term accounts for not to the work of thermodynamic forces, but function cased by the external parameters a. Now let us turn our attention to the analog iltonian systems. The second law of thermodynamics has the dQ Ì h?aî dS ?aî:

the change in the internal energy due rather to change in the distribution of the second law for the non-Hamform ?92î

This implies that there exists a function of state S (a) called entropy. The function h (a) acts as integration factor. Let us prove that (92) follows from the statistical definition of dQ in Eq. (91). For Eq. (91), we take the distribution that is defined by the Hamiltonian, and show that (91) can be reduced to (92). Let us assume that Eq. (11) can be solved in the form H Ì h?aîh?qN î; where h depends on the distribution qN. The function h (a) is a function of the parameters a = {a1, a2, ..., an}. We rewrite (91) in the equivalent form Z dQ Ì ?h?aî h?q?r; p; aî; aî? C ?aîî da qN ?r; p; aî dN r dN p: ?93î New term with C (a), which is added into this equation, is equal to zero because of the normalization condition of the distribution function qN Z C ?aîda qN ?r; p; aî dN r dN p Ì C ?aîda 1 Ì 0: We can write Eq. (93) in the form Z dQ Ì h?aîda K ?qN ?r; p; aîî dN r dN p; where the function K = K (qN) is defined by oK ?qN î Ì h?qN î? C ?aî=h?aî: oqN ?95î

?94î

We see that the expression for dQ is integrable. If we take 1/h (a) for the integration factor, thus identifying h (a) with the analog of absolute temperature, then, using (92) and (94), we can give the statistical definition of entropy ZZ ?96î S ? aî Ì K ?qN ?r; p; aîî dN r dN p ? S 0 : Here, S0 is the contribution to the entropy which does not depend on the variables a, but may depend on the number of particles N in the system. Note that the expression for entropy is equivalent to the mean value of phase density function S ph ?r; p; aî Ì K ?qN ?r; p; aîî=qN ?r; p; aî? C ?aî: ?97î


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Sph is a function of dynamic variables r, p, and the parameters a = {a1, a2, ..., an}. The number N is an arbitrary natural number since we do not use the condition N ) 1 or N fi 1. Note that in the usual equilibrium thermodynamics the function h (a) is a mean value of kinetic energy. In the suggested thermodynamics for the nonHamiltonian systems h (a) is the usual function of the external parameters a = {a1, a2, ..., an}.

8. Conclusion The aim of this paper was the extension of the statistical mechanics of conservative Hamiltonian systems to non-Hamiltonian and dissipative systems. In this paper, we consider a wide class of non-Hamiltonian statistical systems that have (canonical or non-canonical) distributions that are defined by Hamiltonian. This class can be described by the non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is defined by Hamiltonian. The special constraint allows us to derive solution for the distribution function of the system, even in far-from equilibrium situation. These distributions, which are defined by Hamiltonian, can be derived analytically as solutions of the Liouville equation for non-Hamiltonian systems. The suggested class of the non-Hamiltonian systems is characterized by the distribution functions that are determined by the Hamiltonian. The constant temperature systems [2-6,9], the canonical-dissipative systems [12,13], and the Fermi-Bose classical systems [13] are the special cases of suggested class of non-Hamiltonian systems. For the non-Hamiltonian N-particle systems of this class, we can use the analogs of the usual thermodynamics laws. Note that N is an arbitrary natural number since we do not use the condition N ) 1 or N fi 1. This allows us to use the suggested class of non-Hamiltonian systems for the simulation schemes [32] for the molecular dynamics. In the papers [40-42], the quantization of the evolution equations for non-Hamiltonian and dissipative systems was suggested. Using this quantization it is easy to derive the quantum Liouville-von Neumann equations for the N-particle statistical operator of the non-Hamiltonian quantum system [26]. We can derive the canonical and non-canonical statistical operators that are determined by the Hamiltonian [17,18]. The condition for non-Hamiltonian systems can be generalized by the quantization method suggested in [40,41].

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