covariant formulation of non-equilibrium statistical thermodynamics

Z. Physik B 26, 397 -405 (1977) Zeitschri f t

Physik B © by Springer-Verlag 1977

Covariant Formulation of Non-Equilibrium Statistical Thermodynamics

Robert Graham

Universit~it Essen, Gesamthochschule, Fachbereich Physik

Received December 23, 1976

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin- equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal. The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.

1. Introduction

Our concern here will be a phenomenological theory of macroscopic systems which are not necessarily in thermodynamic equilibrium. Systems are thought to be "macroscopic" if there exists a well defined level of description, in which a few "macroscopic" variables are sufficient to specify completely the "state" of the system at each instant of time, making it unnecessary to refer back to earlier times for that purpose. Macroscopic systems are also thought to be large in some sense, so that the macroscopic variables take on continuous values and vary continuously in time. A theory of fluctuations in such systems then, will have to deal with continuous, generally nonlinear, multi-dimensional Markovian random processes. The Fokker Planck equation is the master equation of such p rocesses -and therefore of non-equilibrium thermodynamics. There is a large body of literature on this subject, from which we can only quote a few references, more or less at random, as pars pro toto [ 1 4 ] . Many authors have studied its derivation for

specific cases. Its validity will be taken for granted in this paper: Its symmetry properties will be studied. The Langevin equation

il v = i f ( q ) + gi~(q) ~i(t); (~i(t) aaussian)

(~(t)) = O, (~'(t) ~k(o)) = b,k 6(t), (1.1)

(summation over repeated indicates implied) is prob- ably closest to physical inuition, since it provides direct information about the time dependent paths qV(t), (v = 1 ... n) of the set of macroscopic variables q. Greek indices number the macroscopic variables, while latin indices number the "fluctuating forces", v = 1 ... n; i = 1 ... m. We will assume below that m > n, for reasons which become clear later, f*(q) and gi~(q) are "state functions" of q taken at the same time as q~. In spite of its intuitive appeal (1.1) is beset with mathematical difficulties, glV(t) is, for example, not well defined, since ~(t) is not an ordinary analytical quantity. From a practical point of view it seems best to imagine q~(t), which is continuous in t but not differentiable, to be approximated by a sequence of

398 R. Graham: Covariant Formulat ion

continuous differentiable ~(t) and to treat 4 v like the sequence of ~ in the sense of a generalized function. This point of view has the advantage of being close to the physical reality, in which an object like O ~ arises just in this way. Mathematically it means, that 0 ~ obeys the formal rules of calculus*. Stratonovich [5] shows how to convert (1) into a Fokker Planck equation. The following short derivation differs from his and is given here because of its simplicity [11]. For every ~ there is a probability density Pc(q t) for observing (q, dq). It satisfies the continuity equation

opc & 4-~;q~ O~P¢=0 (1.2)

where 0 * is given by (1.1). The formal solution of (1.2) (T= time ordering operator)

P¢(q t)= r exp dT, ~qV(z) ff(q(T,))

Oq~(T, ) g~V(q(T,)) ~(T,) P(q, O) (1.3)

can be expanded in the form

Pc(q,t)=T exp dT, ~q~(T,) ff(q(T,)) ,=o n[

t

• ~d'c I . . . dT,. 0 ~? o Oq,~(T,1) g~i~ ... Oq,~,(T,,,)

Vn i 1 1 " g i~{ (72) . . . { in(T'n) P ( q , O ) (1.4) )

and then be averaged over {. (Pc(q, t)) reduces to P(q, t), the probability density to observe q at time t. The Gaussian property and the &correlation of the allows to do half of the time integrals in Equation (1.4) upon averaging**. 1 ~ I n \ \ Observing that for each n the re are n[/{~i~},2 "/2)

equal terms and taking m = n/2, since only terms with even n survive the averaging, we are left with

P(q, t)= T exp dT, Oq~(T, ) ff(q(z))

1 m % i

= = o o ~q~(~;)

• gzV ~q~'(5) gku 6'~ P(q' O) (1.5)

* In more technical terms: We interpret Equat ion (1.1) as a stochastic equation in the sense of Stratonovich, not in the sense of Ito cf. [10]

** The averaging over ~ has to be done with fixed q. However, since no backreaction of the q on the ~ is assumed, the statistics of the ~ is not altered by this constraint; averages like (gi~(q)~i(z)) vanish for fixed q

which can be resummed to the form

p(q, t )=T{exp[idT, ( -Oq~(r~ f f (q(T'))

+~6 ~ gz ~ OqU(T,) gk" P(q,O). (1.6)

Taking the time derivative of (1.6) we arrive at the Fokker Planck equation

~P(q,t)~t f ~ 1 02 t ~-~q K~(q)-~ 2 0qVOq ~ Q~(q)~P(q,t) (1.7)

with

1 ~g~(q) gk~,(q) 6i k (1•8) K*(q) =if(q) + 2 Oq ~'

Q*~'(q) = g~V(q) gkU(q) 6 ik. (1.9)

Equation (1.7) is mathematically much safer ground than (1.1). Also from a physical point of view it is a better starting point for non-equilibrium thermodynamics since it does not contain the ~z(t), which are remnants of microscopic variables and belong to a completely different phase space than the q. (Hence our use of latin indices to count them.) The forms of (1.1) and (1.7) change under general coordinate transformations q'=q'(q). Physical properties are independent of the coordinates used. Hence they must be formulated in terms of covariant quantities. The study of the transformation behaviour of (1.1) in Section 2 and of (1.7) in Section 3 will allow us to identify such covariant quantities.

2. Covariance o f the Langevin Equat ion

Geometrical concepts are introduced into our considerations by looking on the coordinate differential dq v as a contravariant vector*• Under a general transformation we have

d,V ~q'~ q =~q~ dq". (2.1)

In the transformed coordinates (1.1) takes the form

(1,~ = f,V(q,) + g,i~ (q,) ~i(t) (2.2)

with

• ,v ~q'~ ,v Oq '~ ~ Oq'V q =~@-q, ~", f = ~ - q , f , g'~= ~q~ g~. (2.3)

The quantities 0 v and fv are thereby identified as contravariant vectors. The matrix gi ~ according to (2.3) does not transform like a tensor. Rather, for

* The notions used here are explained, e.g. in the introductory chapters of any textbook on general relativity. For a particularly nice one cf. [12]

R. Graham: Covariant Formulation 399

each value of the index i ( i= 1 ... m), g~ transforms like a contravariant vector, too. It defines, therefore, a set of contravariant vector fields in phase space. Finally, {~ in (1.1) is left untouched by the transfor- m a t i o n - i t is a quantity in a different linear space whose properties we need not consider in a macroscopic theory. It is remarkable that the Langevin equation, with its intuitive physical appeal, is already formulated from the start in manifestly covariant form, once g~ has been recognized as having its two "legs" in two different spaces. A contravariant tensor in phase space can be constructed from g~ by

Q~U(q) = g~(q) gkU(q) 6,k. (2.4)

Q~U is non-negative, by construction. We assume now and in the following that it is non-singular as well i.e. Q=llQ~U]l@O. Note that a necessary condition for Q 4= 0 is m > n. Q~U can then be used as a metric tensor in phase space. Its inverse is written as Q~u,

Qa, Q,~ = Q~u QUa = 6f (2.5)

and transforms like a covariant tensor. Q~u is used to lower the indices of contraviariant vectors (and tensors) and to transform them into covariant ones. In particular we can define the set of m covariant vector fields g~ (i= 1 ... m) by lowering the index of gi v

- ~ (2.6) giv : Q~ gi u ; glv gk u t~ik : gi u gkv tSik -- (~v

g~ has the interesting property that its contraction with a contravariant vector gives a set of m scalars. Thus, we may write the Langevin equation not only as one vector equation but also as a set of m scalar equations in the form

dz~ = (f~ + g~k {k) dt (2.7)

with

dz~=gi~dq~; f~=g,, f~; g,k=&~ gk L (2.8)

The dz~ are not complete differentials, in general, and thus furnish a set of anholonomous coordinates. Note, that the dz~ are scalars in the phase space of the qL Equations (2.7) remain invariant under a general change of coordinates q' =q'(q). By their construction it is clear that only n of the m (2.7) are linearly independent. Together with the m - n linear anholonomous relations between the dz~ following from dz~=&~dq ~ after elimination of the dq ~ they determine the dynamics of the system uniquely. The dz~ are holonomous coordinates only in the special case where the holonomity condition

=Og~ _ S g u = 0 (2.9) ~i~u- 8q" 8q ~

is satisfied.

Note, that the left hand side of (2.9), for each value of i, transforms like a covariant tensor, a property which is not shared by its two pieces separately. Hence holonomity is an invariant physical property, independent of the special coordinates used. If it is satisfied, the use of (2.7) instead of (1.1) may be advan- tageous. In particular, if m = n, Q + 0 implies ]1 gkull 4= 0 and gik=gi~gk~=6ik . In this case it is sufficient to consider the phase space of the z~ with Euclidean metric. The connection of Z~v u with the curvature of phase space will be considered below.

3. Covariant F o r m of the Fokker Planck Equation*

Let us now introduce a general coordinate transformation q' =q'(q) into (1.7). The new equation

8P' { 8 1 8 2 } 8t = - ~ K'~-~ 2 8q'~Sq 'u Q'~u P' (3.1)

is obtained in terms of the quantities P', K 'v, Q'~. The connection of K '~, Q'~" with the unprimed quantities is most easily read off (2.3) and (1.8), (1.9).

,~ 1 dg'i~ ,u~,k 8q '~ KU+ 1 QUa 02q '~ K'~ = f + 2 8q'*' gk 0 = Sq ~ 8q. Sqa

8q '~ 8q 'u Qa~. (3.2) Q,~U 8q a c3q ~

In order to connect P' with P it is best to recall that P dq=P'dq' is a probability, i.e. a dimensionless invariant number which must transform like a scalar.

dq'= ~qqSq' dq (3.3)

transforms like a scalar density of degree - 1, i.e. like

1/~, where Q = dlQ~Ul]. The probability density transforms like a scalar density of degree + 1, accordingly,

i.e. P ] / ~ is a scalar. Equation (3.2) repeats our earlier information on the properties of Q~u which appears as the "diffusion matrix" of the Fokker Planck equation. At the same time it shows that the "drift vector" of the Fokker Planck equation does not, in fact, transform like a contravariant vector. However, it is possible to ex- tract from K ~ that part which does not transform properly by noting that

8 Q'~ ½ ] / ~ 0q'U ] / -~

- 8qa 8qU 1//~] " 2 ~5 8 q ~ q 2 (3.4)

* Some of the covariance properties have already been used in the literature, cf. [2]

400 R. Graham: Covariant Formulation

produces the same "inhomogenity" in the transformation law as K ~. Hence

Q~. h ~ = K ~ - ½ I / Q ~3q, l~ ~ (3.5)

transforms like a proper contravariant vector. It is now interesting to look back at the Langevin equation and try to express the contravariant vector f~, also associated to the drift through phase space, in terms of the contravariant drift vector h v of the Fokker Planck equation. We obtain at once

h ~ = f~ + 1 ~ (3.6)

where l ~ necessarily is another contravariant vector, given by

v 1 l - ~ QVg gl ~ Ekg ~ 6 ~k = Qvg Ig. (3.7)

Comparison of (3.7) and (2.9) shows that l~= 0 only in systems which have the property of holonomity. In such system there is no difference of the contravariant drift vectors of Langevin equation and Fokker Hanck equation. We now want to formulate the Fokker Planck equation in a manifestly covariant form. To this end we need to specify a covariant derivative of a tensor field

0V g~ Vg~;~ = - - - + FU=~ V z~ + F ~ V g~ (3.8) 0q ~

where F "~ is the affine connection of the Riemann geometry in phase space, defined by the metric tensor Qg~. Requiring

the affine connection is given by the usual expression, symmetric in #, v

1 (3.9) F , = ~ Q ~ [ ~ _ ~Q=~ oQg~] ~qV c3q ~ ]

or, in terms of g[, by

~gkg a IPJ'g v = ~Sik gi'a" ~ "1- CO .v

O'Zgv= 2 "el t3,~ ~Aik'{giv "kKU~ + gin SkKv -- gi~: ~kgv)" (3. l 0)

The last expression clearly exhibits the contribution to F)'g~ due to anholonomity, coxg, is a proper tensor.

Covariant derivatives of general contravariant tensors are defined by treating all contravariant indices like #, v in (3.8). The covariant divergence is

c3 V g V U, g = l// Q ~3 q, l /~ Q (3.11)

while the covariant gradient of a scalar ~ is simply

We can now define a scalar probability density S by

"S(q t) = P(q t) F/Q (3.13)

and an invariant volume element

d f2=dq / ] /~ (3.14)

so that P d q = S d f 2 . Expressing K ~ in terms of h ~, QVU, and ordinary derivatives in terms of covariant derivatives we read- ily obtain the covariant form of the Fokker Planck equation

S=- - [ h ~ S - ½Q~gS;g];~ = -g~;~. (3.15)

The probability current g~ in (3.15) is only defined up to a sourceless term d g ~ which we may generally write in the form

v - - v # - - I v # A g - A ;g-~(A S);~ (3.16)

with

A~u= - A g~, A~u= - A g~ (3.17)

both antisymmetric tensors. Equation (3.15) then appears in the form

S = -[(hV+½A~U;g)S-½(Q~U-A~g)S;g];~. (3.18)

Thus, the covariant drift vector h v is uniquely defined only if we require the tensor in front of the S;g-term in (3.15), (3.18) to be symmetric, which implies A TM

=0. This convention will be used in the following (cf. Section 8, however).

4. Harmonic Coordinates

From a physical point of view there is something strange to the result (3.5). It states that there is a difference between the quantity K ~, related to the vector O v by

(O v) = ( K v) (4.1)

and the drift vector h v, which appears in the covariant Fokker Planck equation. In macroscopic systems, to which thermodynamics applies, Equation (3.5) seems strange, since there it violates fundamen- tal homogeneity properties. Typically, in thermodynamic descriptions qV and c) ~ are extensive quantities proportional to the size V of the system. The same should be true for K ~ and h ~. However, QVU in macroscopic systems is also proportional to the size of the system, i.e. its derivative with respect to q~ is independent of the size. We have

R. Graham: Covariant Formulation

to conclude, then, that h ~ and K v in (3.5) cannot both be extensive quantities, unless

a (QVU/]/~) = 0. (4.2) aq"

As a matter of fact, Equation (4.2) can always be satisfied by an appropriate choice of coordinates (cf. e.g. [12]), usually called harmonic. Equation (4.2) is not generally covariant, however. It rather plays the role of a gauge condition, which breaks general covariance. In order to satisfy (4.2) we assume first that

Fv - - - - I /Q 0 ~ 4 =0, (4.3)

introduce a general coordinate transformation q '= q'(q), and require (cf. Eq. (3.4))

F'*= _] / /~ 0 Q,VU_Oq,V F a - Q "a ~2q,V =0. (4.4) c3q'U V ~ c~qZ Oq~ c~q~

Equation (4.4) is an elliptic second order partial differential equation for the harmonic coordinates as a function of the original coordinates. Defining the n-dimensional Laplacian

- = (QV.

= QVU qS. (4.5) Oq v gq" 3q v]

Equation (4.4) takes the form

A(") q'v=o. (4.6)

It may be useful to express the thermodynamic gauge condition (4.3) also in terms of gi L Then it takes the form of a condition for the vector of anholonomity (3.6)

i v =1 g., cgk" 6ik. (4.8) 2 , 0q"

In particular, if I v=0, the right hand side of (4.8) vanishes for harmonic coordinates. Equation (1.8) is reduced to

K v =f~ + I v = h ~ (4.9)

in the harmonic gauge. In systems with vanishing anholonomity, l~=0, described in harmonic coordinates, there is no difference between the three quantities f v, K v, h v.

5. Covariant Evolution Criteriom

Well known evolution criteria are associated to the Fokker Planck equation (cf. e.g. [6]), which we may state in covariant form. An invariant time dependent

401

potential 0 is defined by the invariant probability density S,

S(q, t) = exp( - 0(q, t)). (5.1)

In terms of 0(q, t) the probability current

g" = h u S - ! t~u ~ (5.2)

appearing in (3.15) may be written as

gU = r uS; r u = h u _ d u; d u = 1 - ~ 0 ; , - (5.3)

We have thereby defined yet another contravariant drift vector r ~, which depends on the particular state of the system (i.e. on 0 (q, t)) and describes the stream- ing velocity of the flow of ensemble points through phase space. If we specialize (5.1), (5.2) for the steady state of the system,

0(q, t)= 0o(q); rU(q, t)= roU(q); au(q, t)=doU(q)

we obtain quantities which are independent of special initial conditions and therefore describe intrinsic properties of the system. We have

So = e x p ( - 0 o ) ; hU=roU+do u (5.4)

do u = 1 -~0o;u , [ro~ exp( -0o) ] ; , =0.

Equation (5.4) implies

1 v v ~ro ;~+ro do~=0.

The average ( 0 - 0 o ) o in the state described by S = e x p ( - 0 ) is always negative and increasing in time. It is negative, since

dq e- 0 _ _e(O_0o))<0. ( 0 - 00)0 = ~ ~ (0 0o + 1 (5.5)

Its increase in time is made obvious by the re- arrangements

d ( ~dq [OOe_O+(O_ 0o) Oe-O] ?7 0 - 0 ° > ° = I / Q \at !

dq =_5 (0_l_0o)(rve

dq = f ~ [(O-Oo);~rV e - ° - ( roV e-*°);ve-(°-°°)]" (5.6)

The last term in (5.6) is obviously zero because of (5.4) and has been added for convenience. Partial integration of the last term and use of Z- roY=½(0 - 0 o ) ;~ leaves us with

d 1 27<0-0°>°

= 2((d ~-dov)(d~- dov)) * = 2((r ~ - roV)(r~- roy)) > 0

(5.7)


which is positive due to the properties of the metric tensor. The covariant vector

) d . - d o u = - ~ ( ~ - ~ o ;,

may be viewed as a "force" which drives the system towards the steady state. It is remarkable that a potential of this force exists; it must be realized, however, that this potential is time dependent; it depends both on the steady state, and on the state at time t*. Equations (5.4), (5.6) therefore ensure the existence of a unique steady state (under the assump- tions implicit in their derivation); however, they are of little help in finding e.g. ~o in the steady state. We list a number of usefull properties of the vectors r", d", roU , do" which are easily proven using (5.2) (5.6):

<4">, = <r"> o = < K"> o (5 .8 )

<(d ~ - do ~) ro~>~ , = 0 = <(r ~ - ro ~) r o ~>o (5.9)

<d~r > 0 - i v - - g < r ;~>q, (5.10)

<r ~ r~>, > <ro ~ ro~>, (5.11)

<h~>q,o = 0 = <do">Oo = <roU>o o (5.12)

- - % / r v X • <ro v d 0 v > o o = 2 \ o ;v/~ko,

<h v h~>oo < (ro ~ ro~>Oo + (do ~ do~>oo (5.13)

<doUdo,>oo = _ l < d o l , >Oo= 1 , _ - ~ < h ;,>¢o>_0. (5.14)

The last inequality (5.14) places a necessary condition on the drift vector h". In harmonic coordinates h ~ = K " ; a vector field hU=K" with too many positive sources in phase space (e.g. K " ; , > 0 everywhere) would not be compatible with (5.14); i.e. such a drift would never lead to a time independent steady state distribution.

6. Lagrangian of Non-equilibrium Thermodynamics

In recent papers [15, 16] the path integral representation of the solution of the Fokker Planck equation (3.1) has been derived. The covariant formulation presented here could there be used with advantage in order to simplify and interpret the final result. If S(qtlq'O) is the invariant conditional probability density, which reduces to Q ] / ~ 6(q -q ' ) for t =0, the result of [16] takes the form

S(qtlq'O)= ~ Dp([q])exp - drL(iM),q(z)) (6.1) q(o) = q'

* The necessity to consider such potentials in non-equilibrium thermodynamics was pointed out by Glansdorff and Prigogine [13]. Our derivation here is similar to one given by Schlggl [14]

with the scalar measure of integration

N-1 t D#([q])=lim 1~[ df2(q(tj)); t~=je; N = - (6.2)

e~O j = l

and the scalar Lagrangian

L(0, q)=l(4~ ~ • 1 v 1 - h ) ( G - h , ) + g h ; ~ + ~ R (6.3)

R in (6.3) is the Riemannian curvature scalar, which is most easily expressed in terms of the affine connection (3.9)

R=Q~x {c~FL~ 3rX ~ ~ x ) (6.4)

In harmonic coordinates L simplifies somewhat

L 1,.~ K . . . . _ K . ) + } } / ~ 0 K" - a¢ (6 .5 )

with the curvatur scalar

6 2 In Q Rh=F;~ru Oq z F ~ ~ vr-- hlt 2-% Oq~#qU"

In systems with vanishing anholonomity R vanishes. This can be seen best by expressing the curvature scalar R in terms of the anholonomous part ogau~ of F~.~ in the following way

R = Q~(couxu ;~ - co,~;, + co,~ co~z, - co, x, o0~z~). (6.7)

In the derivation of (6.7) we have assumed that we have m = n in (2.6) and we made use of gi~g[=f ik . Equation (6.1), in our phenomenological description, is the analogue of the partition integral of equilibrium statistics. The description furnished by the Lagrangian L is somewhat intermediate between a full microscopic description and a completely de- terministic macroscopic one; L in our theory has the status of, e.g., a Ginzburg Landau expanded phenomenological "Hamiltonian" or "free energy" in phenomenological statistical theories of thermodynamic equilibrium. L contains the full information about the system. In particular

t~2L 0L Q~u=aO~ Ut, ; h ~ - a(t~ 4-g1~. (6.8)

Like in equilibrium statistics the evaluation of the "partition integral" in (6.1) has to be done by per- turbative methods. In macroscopic systems saddle point methods and loop expansions are most appropriate. The approximate theory, generated in this way will be considered elsewhere.

7. Covariant Potential Conditions

Sometimes the process considered has a certain symmetry, which involves the reversal of time. In order to


describe that symmetry we define an operator To by

ToF(q,r,2)=F(~l, -%)7); To2=l (7.1)

for any F. )o={)J, ~ = l . . . p } is a set of external parameters (e.g. magnetic field). In particular

T o Z = - - r ; Toq~=gl~=t~uqU ;

r o )J =)~ = s~ 2" (7.2)

with

t~u tu~ = 6~; s~ s~ = 6~. (7.3)

If the system in its steady state has the symmetry described by To, it may be called time reversal symmetric, since T O involves the reversal of time. In particular, such systems satisfy

T O So(q, 2) = So(q, 2)

r o S(q~lq'O, 2)=S(qrlq'O, 2). (7.4)

If (7.4) holds, the system is also said to be in detailed balance withrespect to the transformation T o defined by (7.1-7.3). It has been shown in [17] that the detailed balance conditions (7.4) are equivalent to a set of "potential conditions" on the quantities K *, Q,U appearing in the Fokker Planck equation. (For the special case where t~u = 6~, = s~, this has already been considered by Kolmogoroff [2], even in covariant form.) Since (7.4) express a covariant physical property, the potential conditions also must hold inde- pendently of the coordinates used. In fact, using the covariant quantities introduced in Section 5, the result of [17], (cf. also [6]) takes the form

do~(q, 2) -±(h*td-2, t ,2)+t*u hU(~, i))

ro~(q, 2) = ½ (h~(q, 2) - tvu hU(~, )))

Q~U(q, 2) = t~z tu~ Q~Z(gl, f~) (7.5)

where do ~ and ro ~ are defined by (5.3). As a con- sequence of (7.5) we have

To do~ = flu do"

Toro v = _ tvu roU

T O Q*" = t~o tu~ Qz~. (7.6)

The symmetry with respect to T o can therefore be used to determine do ~ from (7.5). Equation (5.4) can then be integrated to obtain ~o. The condition of integrability ("potential condition") following from (7.5) and (5.4) can be written in the covariant form

(h~(q, 2) + t~u hU(~, ~));~ = (h~(q, 2) + t~, hU(~, fo);~. (7.6)

By choosing the coordinates and the external parameters appropriately we can always achieve that t*u

and s~p are both diagonal; the eigenvalues are _+ 1 because of T 2 = 1.

8. Coupling of Macroscopic External Forces

If external forces are coupled to one of the systems which we consider it will generally respond in all its degrees of freedom, microscopic and macroscopic. The investigation of that response therefore requires the full microscopic theory. However, by cleverly chosing the way in which the external forces are coupled to a macroscopic system we might be able to avoid a separate response of the microscopic degrees of freedom and couple only to the macroscopic variables. It is well known that and how this aim can be achieved, if the system is in thermodynamic equilibrium [18]. It is equally well known [18], that in thermodynamic equilibrium a "macroscopic" fluctuation dissipation theorem holds for the linear response to such macroscopic external forces. Here, we will consider this question for systems in non-equilibrium steady states, using the covariant language developed above. We begin by noting, that in the steady state the probability current go ~ is conserved (cf. Eqs. (5.4))

- ~ r ~o-0o~ - 0 . (8.1) g o ; v - -k 0 ~ / ; v -

The conserved current go ~ is an intrinsic property of the system in the steady state. Assuming that it is known, we may use it as a source for constructing an antisymmetric tensor F~U= - F u~ by the requirement

go ~ = F~U;u" (8.2)

Equations (8.2) are of the same form as the in- homogeneous Maxwell equations in electrodynamics. The tensor F TM is analogous to the tensor of field strengths in electrodynamics. The analogy makes obvious, that indeed (8.2) can be satisfied but does not yet determine the "field strength" F TM uniquely. We may, e.g., require that the homogenous "Maxwell equations"

F~u; ~ + F~u;~ + F~a; u = 0 (8.3)

are also satisfied, which then guarantee [19] that

F~, = ~b~; u - ~b,, ~ (8.4)

can be derived from a vector potential 4~, which is determined by go" up to a gauge transformation.* From (8.2) we obtain

ro~= go~ eOO= ½ eC, O[A~U e-q,o] -±tAbU _ A~U,I, A ; U - - 2 k ;U 'g0;U!

(8.5)

* Sometimes it may be more desirable to use the freedom in F TM in order to simplify A TM in (8.6). The special class of processes with QVU, A ~, both independent of q has been studied by F. Haake (unpublished)


where

A~" = F~" e *° (8.6)

is also an antisymmetric tensor, A~U=-A ~, which can be calculated, in principle, from (8.2)-(8.6). It is useful to know, that the representation (8.5) of ro * in terms of an antisymmetric tensor A TM exists. We use this knowledge to cast the Fokker Planck equation into a new form. First we note that with (5.4)

h ~ = _ ½ (A,U + Q~u) ~o;, + ½A~U;, • (8.7)

Taking now A~u= - A TM in (3.18) we obtain the new form

= ½ (Kv'(0 o;, S + S;,));~ (8.8)

with K ~u = A TM + Q~,. Equation (8.8) makes it obvious that

S o = exp ( - ~o)

is its steady state solution. Generalizing methods of thermodynamic equilibrium [18], we now define the macroscopic external force Fv which is conjugate to the variable q~ as a q- independent force whose effect on the steady state distribution S o = e x p ( - O o ) i s to change ~o into ~o -F~q ~. Equation (8.8) makes it obvious how such a force has to enter the Fokker Planck equation.

= ½ [K~U(0o;u S + S;u)] ;, -½ [K'UF, S];,. (8.9)

It is clear that our definition of F~ is not covariant, since a general coordinate transformation of (8.9) will transform F~ into a q-dependent quantity, which will then not be the macroscopic external force conjugate to q'*. This is as it should be. In an external magnetic field, e.g., any choice of coordinates with the magnetization as an independent variable is cer- tainly priviledged. We will consider (8.9) for time dependent F, also*. The response function R~u(r), defined by

R'u(~) 3<q~(z)) (8.10) aF,(0) ~=0

is easily calculated by first order perturbation theory.

R~U(z)= ½ 0('c) aK%\ "{q~(z) (g au /ao°(~÷'~l alnQ)-~qx • (8.11)

In particular, we have

R~"(0 +) = ½(K~").

Using A "z

Kzu=QXU+AaU; Aux,z=~/Q aq~ V/Q

• The time dependence is, of course, restricted to the macroscopic time scale

which holds because of A ) ' " = - A ~, we can rewrite (8.11) in the form

R~"(~)=½ 0(~) (qV(z) ((Q~" + AZ") ~o;~

aq I / (8.12)

With (8.5) and (5.4) we have

R~U(~)=O(~) lq'(z) (ro"-do~'-½ ]/Q ~(Q";'/]/-Q) ~ (8.13)

Equations (8.9) and (8.13) make it clear, that do u and ro u must be known before the coupling of macroscopic forces to the system and the linear response can be determined. This knowledge is easily obtained, if the system has a detailed balance property. It is usually very difficult to get if no such symmetry exists. Practically all applications have dealt with the former case. In addition, the integration of (8.2) is necessary if the perturbation of the Fokker Planck equation due to the macroscopic forces is to be determined. The latter task is particularly simple in systems with detailed balance, if ro ~ is derived from ~o as a Hamiltonian (cf. [20] for an application in that case).

9. Fluctuation-Dissipation Theorem

The two point correlation function

C,U(O = (qV(r) qU) = {qV(0) qU(- ~)) = CU~(- ~) (9.1)

in the steady state can be related to the response function. From (9.1) we have

C~U(-c) = - C~*(--c) (9.2)

C~U(~) is easily expressed in terms of the coefficients K ~ and Q~" of the Fokker Planck equation, which we write in the abbreviated form P = L P , for convenience. Indeed, for

CVU(z)=~dqq~ ee~LqUPo=Sdq(L+ q~)eW'~qUPo . (9.3)

The two forms of (9.3) lead to two expressions

d~(~) ¢0 ~q~ I/

\ g V Ut~) ~ } q~/ (9.4)

= O(~)(K~(~)q") -- O ( - 0 (q~(~) K") . (9.5)


Comparison of (9.4) with (8.13) immediately leads to the fluctuation dissipation theorem.

~C~(r) R~"(r) + RU~(- z). (9.6)

dr

The validity of this theorem only depends on the correct coupling of the macroscopic forces to the system, as given in (819). It is thus valid for any macroscopic system in a non-equilibrium steady state. For systems with a detailed balance property, Equation (9.6) was already given in [20]*. For the Fourier transforms we have

C~(o) =--2 R"~u(09) (9.7) O9

where

tt 1 v R ~"(o9) = ~ (R "(09) - RU~( - 09)) (9.8)

is the absorptive part of the response function. We note the symmetries

R " * u ( c o ) = - - R " " * ( - co)

[R"~"(og)]* = - R"~"( _ co) = R,,~(Og) (9.9)

which shows that R"~(og) is just the hermitian part of R~U(og). For systems with detailed balance, we have, in addition

R"*u(og) = e* eu R"U~(og) (9.10)

in a representation where t~ = e * 6~ in (7.3). It is clear, that the usual analyticity properties and Kramers-Kronig relations hold for the response functions. The static response

6<q~> =z~UFu (9.11)

to a static force is given by

do9 do9 R'~"(og) _ ~ ~ C~.(o9 ) = (q~ q.> (9.12)

like in usual equilibrium theory. In summary the results of Sections 8 and 9 are the following: After appropriately coupling the external forces to the system, we recover all properties of the linear response which are familiar from systems in thermodynamic equilibrium, except of the symmetry properties under time reversal. The latter is therefore of no principal importance to the fluctuation dissipation theorem. Our conclusion, here, is in accord with earlier work for chemical non-equilibrium sys-

• For other special cases cf. [6, 8, 21 23]; even more general cases than here are considered in [9]

tems [24]. However, the practical importance of a time reversal symmetry, if it is present, is overwhelm- ing. In fact, it gives the only general clue to the correct coupling of the forces to the system apart from explicitely solving the Fokker Planck equation for the steady state probability current and integrat- ing (8.2), eventually with the constraints (8.3). Although our considerations in Section 8 have demonstrated that the latter approach is possible if detailed balance is absent, its feasibility in a practically important problem remains to be shown.

It is a pleasure to thank Fritz Haake for useful conversations.

R e f e r e n c e s

1. Kolmogoroff, A.: Math. Annalen 108, 149 (1933) 2. Kolmogoroff, A.: Math. Annalen 108, 766 (1933) 3. De Groot, S.R., Mazur, P.: Nonequilibrium thermodynamics.

Amsterdam: North Holland 1962 4. Green, M.S.: J. Chem. Phys. 20, 1281 (1952) 5. Stratonovich, R. L.: Topics in the Theory of Random Noise,

Vol. 1. NewYork: Gordon and Breach 1963 6. Graham, R.: Springer Tracts in Modern Physics, Vol. 66, 1.

NewYork: Springer 1973 7. Haken, H.: Rev. Mod. Phys. 47, 67 (1975) 8. Enz, C.P.: Lecture Notes in Physics, Vol. 54, 79. NewYork:

Springer 1976 9. H~inggi, P., Thomas, H.: Physics Letters C, to be published

10. cf. Stratonovich, R.L.: Conditional Markov Processes and their Application to the Theory of Optimal Controll. NewYork: Elsevier 1968

11. Haake, F.: private communication 12. Weinberg, S.: Gravitation and Cosmology. NewYork: John

Wiley 1972 13. Prigogine, I., Glansdorff, P.: Physica 31, 1242 (1965) 14. Schl/Sgl, F.: Z. Physik 243, 309 (1971); 244, 199 (1971) 15. Graham, R.: Phys. Lett. 38, 51 (1977) 16. Graham, R.: Z. Physik B 26, 281 (1977) 17. Graham, R., Haken, H.: Z: Physik 243, 289 (1971); 245, 141

(1971) 18. Kadanoff, L.P., Martin, P.C.: Ann. Phys. (N.Y.) 24, 419 (1963) 19. Adler, RJ. , Bazin, M.J., Schiffer, M.: Introduction to General

Relativity. New York: McGraw Hill 1965 20. Ma, S., Mazenko, G.F.: Phys. Rev. B 11, 4078 (1975) 21. Agarwal, G.S.: Z. Physik 252, 25 (1972) 22. Deker, U., Haake, F.: Phys. Rev. A 11, 2043 (1975) 23. Enz, Ch.P.: preprint 1976 24. Jiihnig, F., Richter, P.: J. Chem. Phys. 64, 4645 (1976); Ber.

Bunsen Gesellsch., to be published

Robert Graham Fachbereich 7 Physik Universit~it E s s e n - Gesamthochschule Postfach 6843 D-4300 Essen Federal Republic of Germany

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covariant formulation of non-equilibrium statistical thermodynamics

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