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Distribution function for systems far from thermal equilibrium
Volume 44A, number 5
PHYSICS LETTERS
2 July 1973
DISTRIBUTION FUNCTION FOR SYSTEMS FAR FROM THERMAL EQUILIBRIUM H. HAKEN Institut für theoretische Physik der Universität Stuttgart, Germany Received 8 May 1973 We present the general stationary solution of the Fokker-Planck equation for a system of interacting subsystems weakly coupled to reservoirs at different temperatures, T 1. For 1) = T, it reduces to the Boltzman distribution function.
In recent years cooperative phenomena in systems far from thermal equilibrium (e.g. lasers) have found an ever increasing interest [1]. Because Fokker-Planck equations are often the adequate means for the theoretical study of these systems, it is highly desirable to have their general solutions available. With respect to stationary solutions two main classes are known: the linear Gaussian process [2] and the solution for systems in detailed balance [3]. In some important systems the corresponding assumptions on the FokkerPlanck equation hold only approximately, however. We therefore started to study systematically the impact of weak perturbations on the general form of the solutions. Wöhrstein [4] has found that if the principle of detailed balance holds and the stationary solution is nondegenerate, it may be extended in the presence of weak perturbations. Here we investigate a different, important case (not confined to detailed balance): The stationary solution be continuously degenerate. We then search for the solution if the system is weakly coupled to reservoirs with different temperatures. We describe the system (which may be also a quantum system described by a Fokker-Planck equation by means of quantum-classical correspondence [5])bytheequation
1= ‘L +L 0
“1~
~
1’’
‘
/ ax1
ji ~x1&’c,-‘
~
g\
1 O’~ /
/
1
n
where gis an arbitrary function of h. After a partial integration, a slight rearrangement of the integration and using that g’(h) is also arbitrary, we obtain our final result: h
fo = N exp
(4)
(X1/X2) dh
where
/
ai~7
~I’~
24(h) =
h = const .i
X(h)
=
f
~
h =const /1
+
I —~-
E
a
i
-~-
~! ax~
\ K11) dx
~—
K ~ ii
...
dXn
~.
...
1
N is the normalization factor. To elucidate the meaning of (4), we consider as special case a system described by a Hamiltonian H = ~ p//2m1 + V(q). We choose: L0 = (a/ax1) G1 and put for! even: x1=q1,G1= aH/ap1,A~=o,K~1=o;forjodd:x1=p1, G1= aH/aq1,M,= 71p1,K11 = ~1171m1kT1. are damping constants, k Boltzmann’s constant and T1 the temperatures of the heatbaths. We obtain —~
—
/
acting in the space ofvariablesx (x1 ,xn). L0 is a linear (differential or integral) operator. We assume, that the solutions of L0 fr 0 are arbitrary functions of a basic function h(x), i.e.f=f(h(x)). We assume in the form L1f ~ M(x)f+ ~ K1. (x)f (2) , ...
the degeneracy of the functionsf, we determine the correct function, f0, in lowest order, by the requirement: (‘ ~ f (h~~ = 0 3
f0
=
N exp
~—
~3H},where j3 =
~ 1
2~7~~k7) / (5)
The new solutions (4) or (5) may then serve as lowest order solutions in a perturbation treatment, which will be discussed elsewhere.
where M and K are small quantities. Because (2) lifts 303
Volume 44A, number 5
PHYSICS LETTERS
References lJ I .g. the articles by R. Graham, S. Grossmann. H. liaken, K. Kawasaki, R. Kubo, R. Landauer, I. Prigogine and P.. Lefever, H. Thomas and others in: Synergetics, ed. II. liaken (Teubner, Stuttgart 1973). [21 Ming Chen Wang and G.E. Uhienbeck, Rev. Mod. Physics. 17 (1945) 323. [31 R. Graham and 11. Flaken, Z. Physik 243 (1971) 289:
304
2 July 197
see also R. Graham in Springer rracts on Physics. to lx published. lor important special cases see R.L. Stratono~ vich, Topics in the theory of random noise (Ness YorL Gordon and Breach, 1963), II. Llakcn. Z. Physik 219 (1 %9i 246. 4 II. Wdhrstein, part ol a thesis, Stuttgart. [5j Haken, Laser Theory, llandbuch der Physik, XXV 2L. (Springer. 1970) pp. 6(1 71 27c 286, where turthcr references arc given.
PHYSICS LETTERS
2 July 1973
DISTRIBUTION FUNCTION FOR SYSTEMS FAR FROM THERMAL EQUILIBRIUM H. HAKEN Institut für theoretische Physik der Universität Stuttgart, Germany Received 8 May 1973 We present the general stationary solution of the Fokker-Planck equation for a system of interacting subsystems weakly coupled to reservoirs at different temperatures, T 1. For 1) = T, it reduces to the Boltzman distribution function.
In recent years cooperative phenomena in systems far from thermal equilibrium (e.g. lasers) have found an ever increasing interest [1]. Because Fokker-Planck equations are often the adequate means for the theoretical study of these systems, it is highly desirable to have their general solutions available. With respect to stationary solutions two main classes are known: the linear Gaussian process [2] and the solution for systems in detailed balance [3]. In some important systems the corresponding assumptions on the FokkerPlanck equation hold only approximately, however. We therefore started to study systematically the impact of weak perturbations on the general form of the solutions. Wöhrstein [4] has found that if the principle of detailed balance holds and the stationary solution is nondegenerate, it may be extended in the presence of weak perturbations. Here we investigate a different, important case (not confined to detailed balance): The stationary solution be continuously degenerate. We then search for the solution if the system is weakly coupled to reservoirs with different temperatures. We describe the system (which may be also a quantum system described by a Fokker-Planck equation by means of quantum-classical correspondence [5])bytheequation
1= ‘L +L 0
“1~
~
1’’
‘
/ ax1
ji ~x1&’c,-‘
~
g\
1 O’~ /
/
1
n
where gis an arbitrary function of h. After a partial integration, a slight rearrangement of the integration and using that g’(h) is also arbitrary, we obtain our final result: h
fo = N exp
(4)
(X1/X2) dh
where
/
ai~7
~I’~
24(h) =
h = const .i
X(h)
=
f
~
h =const /1
+
I —~-
E
a
i
-~-
~! ax~
\ K11) dx
~—
K ~ ii
...
dXn
~.
...
1
N is the normalization factor. To elucidate the meaning of (4), we consider as special case a system described by a Hamiltonian H = ~ p//2m1 + V(q). We choose: L0 = (a/ax1) G1 and put for! even: x1=q1,G1= aH/ap1,A~=o,K~1=o;forjodd:x1=p1, G1= aH/aq1,M,= 71p1,K11 = ~1171m1kT1. are damping constants, k Boltzmann’s constant and T1 the temperatures of the heatbaths. We obtain —~
—
/
acting in the space ofvariablesx (x1 ,xn). L0 is a linear (differential or integral) operator. We assume, that the solutions of L0 fr 0 are arbitrary functions of a basic function h(x), i.e.f=f(h(x)). We assume in the form L1f ~ M(x)f+ ~ K1. (x)f (2) , ...
the degeneracy of the functionsf, we determine the correct function, f0, in lowest order, by the requirement: (‘ ~ f (h~~ = 0 3
f0
=
N exp
~—
~3H},where j3 =
~ 1
2~7~~k7) / (5)
The new solutions (4) or (5) may then serve as lowest order solutions in a perturbation treatment, which will be discussed elsewhere.
where M and K are small quantities. Because (2) lifts 303
Volume 44A, number 5
PHYSICS LETTERS
References lJ I .g. the articles by R. Graham, S. Grossmann. H. liaken, K. Kawasaki, R. Kubo, R. Landauer, I. Prigogine and P.. Lefever, H. Thomas and others in: Synergetics, ed. II. liaken (Teubner, Stuttgart 1973). [21 Ming Chen Wang and G.E. Uhienbeck, Rev. Mod. Physics. 17 (1945) 323. [31 R. Graham and 11. Flaken, Z. Physik 243 (1971) 289:
304
2 July 197
see also R. Graham in Springer rracts on Physics. to lx published. lor important special cases see R.L. Stratono~ vich, Topics in the theory of random noise (Ness YorL Gordon and Breach, 1963), II. Llakcn. Z. Physik 219 (1 %9i 246. 4 II. Wdhrstein, part ol a thesis, Stuttgart. [5j Haken, Laser Theory, llandbuch der Physik, XXV 2L. (Springer. 1970) pp. 6(1 71 27c 286, where turthcr references arc given.