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Mo dern Physics Letters B, Vol. 17, No. 23 (2003) 1219­1226 c World Scientific Publishing Company

CLASSICAL CANONICAL DISTRIBUTION FOR DISSIPATIVE SYSTEMS

VASILY E. TARASOV Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia tarasov@theory.sinp.msu.ru Received 13 August 2003 We derive the canonical distribution as a stationary solution of the Liouville equation for the classical dissipative system. Dissipative classical systems can have stationary states that lo ok like canonical Gibbs distributions. The condition for non-p otential forces which leads to this stationary solution is very simple: the p ower of the non-p otential forces must b e directly prop ortional to the velo city of the Gibbs phase (phase entropy density) change. The example of the canonical distribution for a linear oscillator with friction is considered. PACS Numb er(s): 05.20.-y, 05.30.-d Keywords : Dissipative systems; canonical distribution; Liouville equation.

1. Intro duction The canonical distribution was defined 101 years ago in the book "Elementary principles in statistical mechanics, developed with especial reference to the rational foundation of thermodynamics"1 published in 1902. The canonical distribution function usually can be derived as a stationary solution of the Liouville equation for non-dissipative Hamiltonian N -particle systems with a special set of potential forces.2 ­ 6 In general, classical systems are not Hamiltonian systems and the forces which act on particles are the sum of potential and non-potential forces. The non-potential internal forces for an N -particle system can be connected with nonelastic collisions. 7 Dissipative and non-Hamiltonian systems can have the same stationary states as Hamiltonian systems.8 For example, dissipative quantum systems have pure stationary states of linear harmonic oscillators.9 We can assume that canonical distribution exists for classical dissipative systems. In this paper we consider the Liouville equation for classical dissipative and non-Hamiltonian N -particle systems. This equation is the equation of continuity in 6N -dimensional phase space. We find that the condition for the non-potential forces which leads to the stationary solution of this equation look like the canonical
1219


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V. E. Tarasov

distribution function. This condition is very simple: the power of non-potential forces must be directly proportional to the velocity of the Gibbs phase (phase entropy density) change. Note that the velocity of the phase entropy density change is equal to the velocity of the phase volume change. In Sec. 2, the mathematical background and notations are considered. In this section, we formulate the main conditions for non-potential forces and derive the canonical distribution from an N -particle Liouville equation in the Hamilton picture. In Sec. 3, in the Liouville picture we substitute the canonical distribution function into the Liouville equation for a dissipative system and derive the condition for non-potential forces. In this section, we consider the Maxwell­Boltzmann distribution function for dissipative systems. In Sec. 4, the example of the canonical distribution for linear oscillators with friction is considered. Finally, a short conclusion is given in Sec. 5. 2. Canonical Distribution from the Liouville Equation in the Hamilton Picture Let us consider the N -particle classical system in the Hamilton picture. In general, the equation of motion the ith particle, where i = 1, . . . , N , has the form pi dpi dri =, = Fi , dt m dt where Fi is a resulting force which acts on the ith particle. In the general case, the force is not a potential and we can write Fi = - U +F ri
(n) i

,
(n)

(1)

where U = U (r) is the potential energy of the system, Fi is the sum of nonpotential forces (internal and external) which act on the ith particle. For any classical observable A = A(rt , pt , t), where r = (r1 , . . . , rN ) and p = (p1 , . . . , pN ) in the Hamilton picture, we have dA A = + dt t The Hamiltonian of this system,
N N

i=1

pi A + m ri

N

F
i=1

i

A . pi

(2)

H (r, p) =
i=1

p2 i + U (r) , 2m

(3)

is not a constant along the tra jectory in the 6N -dimensional phase space. From Eqs. (2) and (1) we have dH = dt
N

i=1

pi U + m ri

N

-
i=1

U +F ri

(n) i

pi , m


Classical Canonical Distribution for Dissipative Systems

1221

i.e. dH = dt
N

(F
i=1

(n) i

, vi ) ,

(4)

where vi = pi /m. Therefore, the energy change is equal to a power P of the non(n) potential forces Fi :
N

P (r, p, t) =
i=1

(F

(n) i

, vi ) .

(5)

The N -particle distribution function in the Hamilton picture is normalized using N (rt , pt , t)dN rt dN pt = 1 . (6)

The evolution equation of the function N (rt , pt , t) is the Liouville equation in the Hamilton picture (for the Euler variables) which has the form: dN (rt , pt , t) = -(rt , pt , t)N (rt , pt , t) . (7) dt This equation describes the change of distribution function N along the tra jectory in 6N -dimensional phase space. Here, is defined by
N

(r, p, t) =
i=1

Fi p

(n)

,
i

(8)

and d/dt is a total time derivative (2): d = + dt t
N

i=1

pi + m ri

N

F
i=1

i

. pi

If < 0, then the system is called is a generalized dissipative system. to the velocity of the phase volume Let us define a phase density of

a dissipative system. If = 0, then the system In the Liouville picture the function is equal change.10 entropy by

S (rt , pt , t) = -k ln N (rt , pt , t) . This function is usually called a Gibbs phase. Equation (7) leads to the equation for the Gibbs phase: dS (rt , pt , t) = k (rt , pt , t) . (9) dt Therefore, the function is proportional to the velocity of the phase entropy density (Gibbs phase) change. Let us assume that the power P (rt , pt , t) of the non-potential forces is directly proportional to the velocity of the Gibbs phase (phase density of entropy) change (rt , pt , t): P (rt , pt , t) = k T (rt , pt , t) , (10)


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V. E. Tarasov

with some coefficient T , which is not dependent on (rt , pt , t), i.e. dT /dt = 0. Using Eqs. (4), (5) and (9), assumption (10) can be rewritten in the form: dS (rt , pt , t) dH (rt , pt ) =T . dt dt Since coefficient T is constant, we have d (H (rt , pt ) - T S (rt , pt , t)) = 0 , dt i.e. the value (H - T S ) is a constant along the tra jectory of the system in 6N dimensional phase space. Let us denote this constant value by F . Then we have H (rt , pt ) - T S (rt , pt , t) = F , where dF /dt = 0, i.e. 1 (F - H (rt , pt )) . kT As the result we have a canonical distribution function: 1 N (rt , pt , t) = exp (F - H (rt , pt )) kT in the Hamilton picture. The value F is defined by normalization condition (6). Note that N is an arbitrary natural number since we do not use the condition N 1 or N . ln N (rt , pt , t) = 3. Canonical Distribution in the Liouville Picture Let us consider the Liouville equation for the N -particle distribution function N (r, p, t) in the Liouville picture (for the Lagnangian variables): N + t
N N

=

i=1

pi N + m ri

N

i=1

(Fi N ) = 0 . pi

(11)

In general, the forces Fi are non-potential forces. This equation is the equation of continuity for 6N -dimensional phase space. Substituting the canonical distribution function: 1 N (r, p, t) = exp (F - H (r, p, t)) kT in Eq. (11), we get 1 - kT Since
N

H + t

N

i=1

pi H + m ri

N

i=1

H Fi N + pi

N

i=1

Fi pi

N

= 0.

is not equal to zero, we have H + t
N

i=1

pi H + m ri

N

F
i=1

i

H = kT pi

N

i=1

Fi . pi


Classical Canonical Distribution for Dissipative Systems

1223

If the Hamiltonian H has the form (3), then this equation leads to
N

i=1

pi m

U +F ri

N i

= kT
i=1

Fi . pi

Substituting Eq. (1) in this equation, we get the following condition for non(n) potential forces Fi :
N

i=1

pi ,F m

N (n) i

= kT
i=1

Fi p

(n)

.
i

Using notations (5) and (8), we can rewritte this condition in the form: P (r, p, t) = k T (r, p, t) . As a result we have that the canonical distribution function is a solution of the Liouville equation for dissipative and non-Hamiltonian systems if the power of the non-potential forces is proportional to the velocity of the phase volume change. Let us consider a chain of Bogoliubov equations11,6 for the Liouville equation of the dissipative systems (11) in approximation: 2 (r1 , p1 , r2 , p2 , t) = 1 (r1 , p1 , t)1 (r2 , p2 , t) .
(n) i (n,e) i

(12)

The non-potential forces F in Eq. (1) is a sum of external forces F and (n,i) internal forces Fi . For example, in the case of binary interactions we have
N

F

(n) i

=F

(n,e) i

(ri , pi , t) +
j =1,j =i

F

(n,i) ij

(ri , pi , rj , pj , t) .

In approximation (12) we can define the force F1 = - where F
(n) 1,eff

(U + Ueff ) +F r1

(n) 1

+F

(n) 1,eff

,

(r1 , p1 , t) =

dr2 dp2 1 (r2 , p2 , t)F

(n,i) 12

(r1 , p1 , r2 , p2 , t) ,

Ueff (r1 , p1 , t) =

dr2 dp2 1 (r2 , p2 , t)U (|r2 - r1 |) .

If we consider the 1-particle distribution then Liouville equation (11) in approximation (12) has the form: p1 1 1 + + (F1 1 ) = 0 , t m r1 p1
(n) 1 (n) 1,eff

(13)

where 1 = 1 (r1 , p1 , t). Let us consider a condition for the non-potential forces (p1 , F
(n) 1

+F

(n) 1,eff

) = mk T

(F

+F
1

)

p

.


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V. E. Tarasov

In this case, we can derive the 1-particle distribution function (as in Sec. 3) in the form: 1 (r, p, t) = A exp - 1 kT p2 + U (r) + Ueff (r) . 2m

This is a Maxwell­Bolztmann distribution function. 4. Canonical Distribution for Harmonic Oscillator with Friction Let us consider the N -particle system with a linear friction defined by non-potential forces F
(n) i

= - pi ,

(14)

where i = 1, . . . , N . Note that N is an arbitrary natural number. Substituting Eq. (14) into Eqs. (5) and (8), we get the power P and the Gibbs phase : P =- Condition (10) has the form:
N

m

N

p2 , i
i=1

= - .

i=1

p2 i = kT , m

(15)

i.e. the kinetic energy of the system must be a constant. Note that Eq. (15) has no friction parameter . Condition (15) is a non-holonomic (non-integrable) constraint.12 Let us consider the N -particle system with friction (14) and non-holonomic constraint (17). The equations of motion for this system have the form: pi dri =, dt m dpi U G = - pi - + , dt ri pi
N

(16)

where the function G is defined by G(r, p) = 1 2 p2 - mk T , i
i=1

G(r, p) = 0 .

(17)

Equations (16) with condition (17) define 6N + 1 variables (r, p, ). Let us find the Lagrange multiplier . Substituting Eq. (17) into Eq. (16), we get dpi U = -( - )pi - . dt ri
N N N

(18)

Multiplying both sides of Eq. (18) by pi /m and summing over index i, we obtain d dt p2 i 2m = -( - )
i=1

i=1

p2 i - m

i=1

pi U , m ri

.

(19)


Classical Canonical Distribution for Dissipative Systems

1225

Using dG/dt = 0 and substituting Eq. (15) into Eq. (19), we get
N

0 = -( - )k T -
j =1

pj U , m rj

.

Therefore, the Lagrange multiplier is equal to = 1 mk T
N

pj ,
j =1

U rj

+.

As the result, we have the holonomic system which is equivalent to the nonholonomic system (16) and (17) and defined by pi dri =, dt m dpi 1 = dt mk T
N

pj ,
j =1

U U pi - . rj ri

(20)

Condition (10) or (15) for the classical N -particle system (20) is satisfied. If the time evolution of the N -particle system (16) has non-holonomic constraints (17) or the evolution is defined by Eq. (20), then we have the canonical distribution function in the form: (r, p) = exp 1 kT
N

F-
i=1

p2 i -U 2m

.

For example, the N -particle system with the forces 2 p Fi = kT
N i j =1

(pj , rj ) - m 2 r

i

can have a canonical distribution that look likes the canonical distribution of the linear harmonic oscillator: 1 (r, p) = exp (F - H (r, p)) , kT where
N

H (r, p) =
i=1

p2 i + 2m

N

i=1

2 m 2 ri . 2

5. Conclusion Dissipative and non-Hamiltonian classical systems can have stationary states that look like canonical distributions. The condition for non-potential forces which leads to the canonical distribution function for dissipative systems is very simple: the power of all non-potential forces must be directly proportional to the velocity of the Gibbs phase (phase entropy density) change. In Refs. 13 and 14, the quantization of evolution equations for dissipative and non-Hamiltonian systems was suggested. Using this quantization it is easy to derive


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V. E. Tarasov

quantum Liouville­von Neumann equations for the N -particle matrix density operator of the dissipative quantum system. The condition which leads to the canonical matrix density solution of the Liouville­von Neumann equation can be generalized for the quantum case by the quantization method suggested in Refs. 13 and 14. The canonical distribution for dissipative quantum systems allows one to consider stationary states of dissipative quantum states as an unusual quantum computer. In general, we can consider dissipative quantum systems as quantum computers with mixed states.15 The quantum gates of this computer are general quantum operations, not necessarily unitary. Acknowledgment This work was partially supported by the RFBR Grant No. 02-02-16444. References
1. J. W. Gibbs, Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (Yale University Press, 1902). 2. B. M. Gurevich and Yu. M. Suhov, Commun. Math. Phys. 49, 69 (1976). 3. B. M. Gurevich and Yu. M. Suhov, Commun. Math. Phys. 54, 81 (1977). 4. B. M. Gurevich and Yu. M. Suhov, Commun. Math. Phys. 56, 225 (1977). 5. B. M. Gurevich and Yu. M. Suhov, Commun. Math. Phys. 84, 333 (1982). 6. D. Ya. Petrina, V. I. Gerasimenko and P. V. Malishev, Mathematical Basis of Classical Statistical Mechanics (Naukova dumka, Kiev, 1985). 7. A. Sandulescu and H. Scutaru, Ann. Phys. 173, 277 (1987). 8. V. E. Tarasov, Phys. Lett. A299, 173 (2002). 9. V. E. Tarasov, Phys. Rev. E66, 056116 (2002). 10. V. E. Tarasov, Mathematical Introduction to Quantum Mechanics (MAI, Moscow, 2000). 11. N. N. Bogoliubov, Selected Works, Vol. 2 (Naukova dumka, Kiev, 1970). 12. V. V. Dobronravov, Foundations of Mechanics of Non-Holonomic Systems (Vishaia shkola, Moskow, 1970). 13. V. E. Tarasov, Phys. Lett. A288, 173 (2001). 14. V. E. Tarasov, Moscow Univ. Phys. Bul l. 56, 5 (2001). 15. V. E. Tarasov, J. Phys. A35, 5207 (2002).