PFC/JA-84-15
TRANSPORT THEORY:
MICROSCOPIC REVERSIBILITY
AND SYMMETRY
K. Molvig and K. Hizanidis
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, Massachusetts
June, 1984
02139
TRANSPORT THEORY: MICROSCOPIC REVERSIBILITY
AND SYMMETRY
K. Molvig and K. Hizanidis
Plasma Fusion Center
Massachusetts Institute of Technology
ABSTRACT
The general theory of Fokker-Planck equation symmetries and their relation to derived transport theory symmetries is developed. The property of
microscopic reversibility implies a symmetry in the Fokker-Planck equation for
processes obeying detailed balance. It is shown that this symmetry is not
sufficient to guarantee Onsager reciprocity for the full matrix of transport coefficients.
The general transport matrix has broken symmetry. A partial symmetry can be
identified. The theory is compared to the different formulation given by Onsager.
1
I.
Introduction
Observatians on systems near thermal equilibrium have shown that irreversible processes are described by linear phenomenological laws which relate
the fluxes to the forces by a matrix of transport coefficients. Furthermore, as
suggested by a large body of experimental evidence and running back to Kelvin's
analysis of the thermoelectric effect' in 1854, these transport coefficients possess
a certain set of symmetries - the Onsager reciprocity relations. Of course, no
completely general theoretical proof of these relations can be made. Onsager 2
showed the connection between these relations and microscopic reversibility in
the underlying dynamics, implying their universal validity. His proof, however,
was essentially empirical and based on two key assumptions. First, that the fluxes
of the extensive variables specifying the thermodynamic state depend linearly on
the forces that cause them. Second, that the average regression of fluctuations of
these variables from a given nonequilibrium state obey the same linear law. This
limitation in the proof of the Onsager relations was emphasized in the review by
Casimir 3 , who noted that this second assumption was, "really a new hypothesis
. . . we do not think it can be rigorously proved without referring in some way
or another to kinetic theory."
In short, the preponderance of observational experience and the general
nature of Onsager's empirical proof suggest that all systems governed by linear
force-flux relationships have obey the reciprocity relations. But in the absence of
a first principles proof of these relations one might still ask if there are systems
that have linear force-flux relations without Onsager symmetry. The present
paper will develop a general kinetic theory that implies that such systems do,
in fact, exist. The theory does not contradict Onsager's original arguments,
but simply restricts their applicability. A large portion of the transport matrix
results from reciprocal scattering processes and obeys the Onsager relations
as a consequence of microscopic reversibility. In general, however, there is a
contribution to the transport arising from essentially kinematic flows that do
not fundamentally obey reciprocal relations. Flows of this type (notably the
Ware pinch 4 ) arise in toroidal plasma confinement configurations but do not
occur in the types of systems first studied by Kelvin and toward which Onsager's
arguments were directed.
2
The necessity of separating out the kinematic flows before identifying. acU'4c
symmetric transport fluxes is in principle well known -9. Perhaps the most
familiar example of this occurs in Enskog's 5 derivation of the hydrodynamic
equations from kinetic theory where the transport calculation is performed in a
frame of reference moving with the macroscopic fluid velocity. One does not have
a symmetric transport theory in the laboratory frame. An analogous separation
of the kinematic flows occurs in the theories of Van Kampen 6 , Graham and
Haken 7 , Zwanzig , and Mori. What makes the example in the present paper
interesting is that these kinematic flows turn out to be linearly proportional to
the forces, so that one could consider them as transport fluxes. Indeed, they
would be indistinguishable, empirically. However, when the kinematic flows are
0
included in the transport matrix, the "Onsager" symmetry can be broken' .
In the original formulation of Neoclassical transport theory", no distinction was made between kinematic and collisional fluxes and a symmetry was
identified and termed "Onsager" symmetry that included both. The Neoclassical symmetry, however, is due to a specific mathematical perturbation theory.
It cannot be traced to a physical reciprocity of the basic microscopic scattering
events and is not fundamentally valid.
The practical purpose of this paper is to develop a general formulation
of a transport theory for confined plasmas capable of treating in a common
framework, for a complex magnetic geometry, collisions and turbulence. This
is important because of the dominance, in practise, of turbulent transport.
The general formulation makes it possible to establish which transport matrix
symmetries are fundamentally linked to microscopic reversibility and therefore
are generally valid.
In Section II, a general Fokker-Planck kinetic equation is derived and the
symmetries in this equation that arise from microscopic reversibility are identified.
This kinetic equation is solved, in Section III, by an expansion which generates
the transport equations. In Section V, the symmetries of the transport equations
are displayed. These are compared in Section V to the symmetries that result
from a different type of transport theory based on the regression of fluctuations
about a global thermal equilibrium. This latter type of transport is what Onsager considered. It can be obtained from our Fokker-Planck equation in certain
3
special cases. By finding the conditions for deriving Onsager's results from this
same kinetic theory, it will become quite clear why the relations fail in the more
general case.
4
II.
Kinetic Theory: Generalized Fokker-Planck Equation
A formulation of transport theory for fusion plasmas that does not depend
on the details of the scattering processes but only some very general properties
has already been developed' 2 . This Lagrangian formulation is based upon an
action-space kinetic equation that in Ref. 12 was obtained for a specific scattering process (Coulomb collisions) in the ideal banana regime limit. In this section
we derive a kinetic equation of the same type that is more general with respect
to both scattering process and collisionality. The derivation will link certain
general mathematical properties of the kinetic equation to fundamental physical principles.. In particular the scattering operator, including turbulence, will
be shown to be self-adjoint as a consequence of microscopic reversibility. The
kinematic flows that will make additional contributions to the transport matrix
can be identified and shown to be separate processes whose conjugate parts are
not related by time reversal. The friction coefficient which is associated with
the kinematic flows will be shown to be time reversible (the name "kinematic"
was chosen on the basis of this property).
A general kinetic equation of the Chapman-Kolmogorov type can now
be written for the distribution function, f(X, t) provided that the underlying
scattering processes (collisional and turbulent) can be considered as Markov
processes.
f(X, t) =f
{dX'}P(X, tlX', t')f(X', t'),
(1)
where the conditional probability density P(X, tAX', t') is the probability of a
transition from the state described in a generalized manner by the vector X' at
time t' to the new state described by X at time t. This equation can describe:
kinematic, reversible flows, where P is a delta function along the deterministic
orbit; binary collisions where P will depend on f so that Eq. (2) is nonlinear; and turbulent scattering processes where the statistics of the turbulent
fluctuations appear in the transition probability function. We restrict the class
of turbulent fluctuations somewhat by requiring that they have saturated and
that all the statistical functions relax on a time scale short compared to the
evolution time scale of the one particle distribution function, f. Then P will
5
depend only on f and not on higher statistical functions, so that Eq. (2) is
closed.
We now proceed to the Fokker-Planck limit of Eq. (2), appropriate to the
turbulent and collisional scattering processes found in fusion plasmas, where the
state vector scatters in small relative steps and in a continuous fashion to a good
approximation. Following the usual Fokker-Planck equation derivation procedure we transform Eq. (2) into a differential equation by using the KramersMoyal expansion and subsequently assume that the transition probability funcX - X'; both the the transition
tion, P, is very peaked with respect to AX
probability function, P, and the distribution function f(X, t) are also assumed
to vary slowly with X. Then Eq. (2) reduces to the Fokker-Planck equation,
Of(X, t) == -
[Kv(X)f(X, t)] +1
Xz-V
at
2
2 aXLI8Xlt)
[D"P(X)f(X, t)]
where the summation convention over repeated indices is used.
vector K"(X) and the diffusion tensor D""'(X) are given by
KU(X)
DVU(X)
alim
f{dAX}AX'w(X, AX, e, t)
r
limC'
C'
{dAX}AX'AX1'w(X, AX, e, t)
(2)
(2
The friction
(3)
(4)
where
w(X, AX, E,t)
P(X + AX, t +EIX, t)
(5)
The transition probability, P(X, tIX', t'), considers the system running forward in time from t' to t. It is obtained by computing microscopic trajectories in
each realization of the scattering fields starting at X' at time t', and averaging
the results over the statistical ensemble for the scattering process. If, in each
realization, time is reversed at time t and the system run backward to time t',
one recovers the time reversed initial data. This is a statement of microscopic
6
reversibility. For stationary processes microscopic reversibility implies the following condition for the transition probability
P(X, tX', t')J(X') = P(TX', -t'TX -t)J(X)
(6)
where T is the time reversal operator and J(X') is the Jacobian, or volume
element in the space X. A detailed proof of this relation is given in the Appendix. Equation (6) expresses the physical condition of detailed balance or
equality for forward and inverse rates for each transition. Not all systems obey
detailed balance. If certain types of radiative processes and external forces are
included, the system can still be described by the Fokker-Planck equation, but
the full transition probability will not have the symmetry, Eq. (6), primarily
because stationarity is lost. Nevertheless, we expect that some of the processes,
the scattering processes, to have the detailed balance property, and that the
corresponding piece of the transition probability will obey Eq. (6).
We now write the Fokker-Planck equation in an alternative form to exploit
the symmetry relation, Eq. (6). This follows from expanding
w(X -
-, AX e, t)J(X
-2)
=
w(X, AX, e, t)J(X)
1
Vw(Y, AX, C, t)J(Y)
2
81"'
(7)
+
which is sufficiently accurate for the Fokker-Planck description. Using Eq. (7),
for w(X, AX, e, t), to rewrite the friction vector, K"(X), gives,
KU(X) = J'(X)H"(X) + 2J(X)aX
where,
7
[J(X)D"(X)
(8)
HN(X) =_
f {dX}AX"
lim
w(X-
ZAX
LAX
,AX,E,t)J(X- A)
(9)
2
2
Upon inserting Eq. (8) into Eq. (2) yields the alternate form Fokker-Planck
equation.
af(X, t)
a
ax H(X)f(X, t)]
_
at
+
J(X)D&)A(X)
af
t
)(10)
Equation (10) is a valid form whether or not detailed balance, Eq. (6), holds.
Now one can also define an invariant distribution function (invariant upon any
transformation of the vectors which describes the state of the system) g(X, t) as
follows
X).
g(X, t) = f(X
(11)
Utilizing the invariant distribution function g in Eq. (10), the Fokker-Planck
equation takes the covariant form
g(X, t)
_-
(V(X)H(X)g(X, t)]
at
1 .7
(X X2jag(X,
(a)(X
J(X)D"_(X)
t)
(2
(12)
We now examine the symmetry properties of the vector H, of Eq. (9).
These are based on the time reversal properties of the vector h'(X, e, t), defined
by
8
(13)
H"(X) = lim h"(X, e, t)
The time reversed form of the function h', since J(TX)
h"(TX, -E, -t)
=
-E'
J(X), can be written.
J{dAX}AX
P(TX + --
2
-, -t)J(TX
, -t - e TX -
2
AX).
2
(14)
Using the symmetry relation for the transition probabilities, Eq. (6), and Eq.
(A28) yields
h"(TX, -e, -t)
P(X-
+C
-C J{dAX}AXV
-tX +
T~
2', + )(+T)
f
AA
X
{dAX}AX"Pna(X -
;T
-,
AX
t+
)
(15)
The time reversal transformation rule for the infinitesimals dX" can now be
as
approximated by introducing the matrix e" (in the most cases '=
follows
E~dX
TdX"
(16)
Since upon, time reversal, the difference operator A (final state - initial state)
changes sign, one has, by virtue of the previous aquation
TAXV = -eCAXIh
(17)
Upon change of variables in Eq. (15) and taking into account Eq. (17) and the
properties of the operator T, Eq. (A), yields upon taking the limit e -+ 0+
9
TH"(X) = H"(TX)
=-f"H4(X)
P,.,(X +
+ E" lim -1 f{dX}AX
C-0+
2
, t; X
-
-,
2
t + e)
(18)
One can now introduce the concept of "reversible", R", and "irreversible"
, I", parts of a vector function A"(X) of the space of states as follows
R"(X) = g[A"(X) - E"AM(TX)]
(19)
I"(X) = g[A"(X) + e"AM(TX)].
(20)
As a consequence of Eqs. (19-20) as well as of the identity e'e
Kronecker delta) one has
6 (6 is the
R'(TX) = -eRA(X)
(21)
I"(TX) = eI M(X)
(22)
The differential operators a/aX" transform, upon time reversal, like the infinitesimals
dX" in Eq. (16); therefore, taking into account Eqs. (21-22) and (A5) yields
the following transformation rules for the operators J-18/aX"JR" and J-1a/aX"JI
= -J~1
T J-1 a JR
''(
ax"}
a JI
T(J-1
9X
T ax
=
aJR"
axV
a c JI
(23)
(24)
that is the operator associated with the "reversible" part of the vector A'
transforms like the time variable t. The transformation relation, Eq. (18),
readily implies that for stationary processes (i.e. P., = 0) one has
10
TH"(X) = 1I"(TX) = -E'H'(X)
(25)
that is, the vector H"(X) is a purely "reversible" vector and, therefore, its
associated t erm in the covariant Fokker-Planck equation, Eq. (12), changes sign
upon time reversal. For general processes, of course, the second term of the right
side of Eq. (18) clearly corresponds to the irreversible part of the vector H"(X).
Concluding, we saw that the Fokker Planck equation, given by Eqs. (10) or
(12) consists of two parts. The last term in these equations, which involves the
operator for the scattering processes, turbulent and collisional, is in self-adjoint
form. The term associated with the flow vector H", on the other hand, can
directly drive flows that do not have a reciprocal process and are not-subject
to the Onsager relations. This term, for stationary processes only, transforms
like a reversible quantity and is associated with kinematic flows. If the variables
X' are even under time reversal (i.e. e' = 6') then Eq. (25) implies that,
for stationary processes, the vector H" is identically zero. This form of the
Fokker-Planck equation has been previously derived in a different context and
with a different way by Van Kampen6, who also identified the terms on the right
of Eq. (10) as "reversible" and "irreversible" flows respectively. Van Kampen
then assumed some special properties of the Fokker-Planck coefficients which
lead directly to a type of transport equation. His assumptions, however apply
only for systems which are close to thermal equilibrium; in this paper we are
interested in states far from thermal equilibrium as it happens with confined
plasmas. His transport equations are compared in Sec. IV to those derived
in the following Sec. M. Graham7 also developed a theory which applies for
stationary steady states far from thermal equilibrium.
11
III.
Transport Equations
We consider a magnetic confinement geometry where the equilibrium particle motion in the absence of fluctuations can be described by action-angle variables X = (J,0). In practical terms, this means that the collisionless orbits are
contained to a good approximation (for longer than a collision time). Although
the use of canonical action-angle variables is not necessary it greatly simplifies
the mathematical symmetries needed to establish the reciprocal relations.
For the tokamak example of Ref. 12, the actions are,
J, = irm(b X v) 2 /f
J=
J=
27ir(i)
C
m
dsa
= MAGNETIC MOMENT
PARALLEL INVARIANT
= DRIFT CENTER FLUX COORDINATE
(26)
(27)
(28)
while the conjugate angles 61, 62, and 63, are, respectively, bounce average
gyrophase, bounce phase and drift phase. The associated frequencies, wi
aH/0Ji, are then the bounce averaged gyrofrequency, bounce (or transit) frequency and bounce averaged drift frequency.
Generally, one is interested in systems where the distribution function,
f, evolves on time scales much longer than the correlation time of the basic
statistical processes. Moreover, it often happens that the angular dependence
of f is destroyed at a faster rate than the transport processes, so that one need
work only with
f(J, t) =
d'Of(J, 6, t)
(29)
Thus, in our tokamak example, the 01 dependence of f is phase mixed away
on the gyroperiod time scale, the 02 dependence on a bounce time scale [the
boundary later between trapped and circulating particles is resolved by colli2
sions so that there is a minimum effective transit frequency of Wtmin=wto 7r
12
v"i/ln(256/v*)J. Both angular mixing processes are much faster than the collision frequency for fusion plasmas. The 03 dependence can be eliminated by
axisymmetry 12. Formally, what happens in Eq. (2) is that for time intervals,
It - t'j, exceeding these mixing times, the probability function, P, becomes independent of the angles, implying the same for f. This gives an effective correlation time for action scattering. We can then write a kinetic equation of
Chapman-Kolmogorov type for the distribution function of Eq. (29)
d3 J'Pj(J,t[J', t')f(J', t'),
f(J, t)
(30)
where,
d3 0d'O'P(J, 0, tJJ', o', t'),
Pj(J, tfJ', t') =
(31)
is the angle averaged transition probability. Following the same procedure as in
See I. yields a Fokker-Planck equation of the form
8
Of
-
at
108
Hf + -T
2
8
(2
(32)
,- j f
Dj
where the explicit forms of the kinematic flow vector H and the diffusion tensor
D are
d 3AJJP (J
H(J)=-
D(J)
lim
r
e-.*O+
'
J
-
t+E - -
t)J(J - -
d3 \JAJA JPi(J + AJ, t + E|J, t)
(33)
(34)
f
For stationary processes the detailed balance symmetry property, Eq. (6), reads
PF(J,tJJ', t') = Pj(J', -t'|J, -t)
since the action-angle description is canonical (i.e. J(X' -+
J.
13
(35)
X)
= 1) and T J =
The last term in Eq. (32), which is self-adjoint, describes turbulent and collisional scattering processes in action space. The term associated with kinematic
flows can, through the coefficient, (AJ 3 /At) = 113 ,directly drive radial flows
that do not have, as we already mentioned, a reciprocal process and are not
subject to the Onsager relations.
Binary collisions from like-particle encounters are described by a bilinear
operator with more complicated symmetries that, as expected, also imply reciprocal relations for the transport coefficients". In the interest of simplicity, we refer
the reader to Ref. 12, and will not discuss such processes further here.
The kinetic equation, Eq. (32), can be solved by an expansion about local
thermal equilibrium to generate the transport equations provided that three
general conditions are met:
1.
Radial scattering is slow compared to velocity scattering.
2.
Velocity scattering is dominated by collisions.
3.
Kinematic flows driven by external forces are small compared to velocity
space flows driven by collisions.
Condition (1) implies, among other things, that the radial extent of the collisionless particle trajectories, A, be small compared to the characteristic macroscopic
scale length, a. We assume it is then possible to choose the three actions such
that one of them, J3 , has two properties which allow it to be used as a radial
variable in constructing a transport theory". First, it must be the actual radial
position of a particle, to order A/a, (apart from constant multiplicative factors).
Secondly, the Hamiltonian function, H(J), must depend weakly on this radial
action such that
W1, W2
> H/8J 3
alnH/8J3 <8Inn/8J,
a InT/0J3 ,
(36)
where n and T are the particle density and temperature. These conditions
assure that the radial position does not significantly affect a particle's energy
and eliminates from consideration systems with spatially localized global thermal
equilibrium states. The actions, J, and J2, are the velocity like variables.
14
Usually, in a plasma, these will be taken to be the magnetic moment and the
parallel invariant.
To formalize these condition, we write the kinetic equation (32) in the form
49f
a
~+ ~- [a(J)Vf] = C(f),
(37)
where, C(f) = 3/8J -D -a/aJf,is the scattering operator and V is a coefficient
parameterizing the external forces that drive the kinematic flows.
For example, in the tokamak problem, V is the toroidal loop voltage giving
a force, F = ecqV/27rR, in the ec direction. The instantaneous increment in
(qV/27rR)e, -O/&J, so that the acceleration
action due to this force is, AJ
coefficient becomes,
1
a(J)
The action vector is written, J
((q/27rR)e, -
0
J)K,
(38)
Jjej + J 2 e 2 + J 3 ea, and we employ the
notation,
J_= Jjej + J2 e2 ,
(39)
for the velocity-like part of the action vector.
In this formulation, quantities such as particle number and energy are
expressed as densities per unit of action, J3, and arise as reduced moments such
as n 3 = f dJidJ2 f. Thus, when the distribution f is known, one obtains a
particle transport equation from Eq. (37) by simply integrating out the velocity
like actions. Since J3 is a (Lagrangian) particle label and not the (Eulerian)
spatial position, the density n 3 does not correspond exactly to the spatial density.
This difference arises from the finite orbital widths of the collisionless orbits, and
is small by a factor of order A/a. It will turn out that this distinction is, to the
order one works in transport theory, irrelevant as far as the density is concerned.
However, since kinematic flows within a surface can result from such effects (e.g.,
the banana current), it is necessary to develop to the relationship between the
action space moments and the various spatial densities required.
15
(',
As spatial variables we will use the conventional magnetic flux coordinates,
0, s) discussed in Ref. 12. The volume element is then,
Vs - V# X Vd
3z
= daddO = Bd 3 ,
(40)
and the specific volume, dV/dO, between flux surfaces is
dV/db
=
|ds d =27rf I
(41)
where the line integral covers one circuit of the poloidal circumference. The path
length, ds1, in Eq. (41) is taken to be positive. The flux surface average of any
function, F, can be written
IdsIdOF/B.
dAF/IV4I
(F} EE
(42)
The subscript, 0, distinguishes this from the bounce average, natural to the
Lagrangian representation, and denoted by
(F=
J
d0 2 F =
w2
f
(43)
dsF/w,
where the path element, da, in Eq. (43) has the same sign as w.
Given any distribution f(J, 0) in the action-angle variables, the spatial
density n(z) can be obtained by projection, viz.
n(z) =
f d Jd O6(x - x(J, 0))f(J, 0),
3
with x, (J,0) determined by the transformation x, v
coordinates this becomes, using Eq. (39),
n(z) =
-
(44)
J, 8. In magnetic flux
d 3 Jd 3OB5(s - s(J, 0))6(0 - #(J, 0))6(ik - ,(J, 0))f(J, 0).
(45)
The flux surface average density takes a particularly simple form. We apply Eq.
(267) to Eq. (45). There results,
16
d))fJ, d06(
(n(V))J
-
g-J,
(46)
)
For present purposes, we write the third action as
J
2irqO+
(47)
+ AJ 3 (0).
C3
Thus, to order AJ 3 /J 3 ~ A/a, J 3 = (27rq/c)?k and to second order in A/a for
f independent of 0, the flux surface average density becomes
2-irq dOb
(n(z))o =
WV-
1jf
dV
J-
dJ3
d
c
(48)
so that (n),/n3 is just the specific volume between surfaces J 3 = const. Actually, the local density n(z), not flux surface averaged, and n 3 have the same
relationship although only to first order in A/a.
Now, we also need an expression for the flow conjugate to the external force,
F, that is parameterized by the flux function, V = V(V;). The actual current in
the e, direction is given by
J=
d3 Jd 3 06(x - x(J, 0))qe, - v(J, 9)f(J),
(49)
which in general depends on the position, (8, 0), within the magnetic flux surface
in addition to 0. To obtain a flux function, I = I(0), for this flow, conjugate
to, V = V(O), we multiply J by 1/27rR and dV/dJ 3 , to give an angular current
per unit J3 and then flux surface average. There results
I = - --
22q dq
(J/21R)
= d 3JW2
dsc6
i uJ u~re
fd
s
f- 1dJ W2
C(J3 - Ahs)
o-
qe
r JJI 27J± '0+ rJ
*Vf
17
f
(50)
Expanding in powers of, AJ 3 /J
I=Jd2JW
2
f
~
%-v
3
= A/a, this becomes,
(j , 2iq')
rsq
f2J
+
d
39J
(J'
J, 27rq+
'\l
-.
(51)
Here the first term gives a moment over the bounce averaged velocity in the
e, direction in a flux surface, while the second term, arising from the radial
excursion of the orbits is of the diamagnetic type. We will label these two
contributions as implicit, P, and explicit, P, respectively. To be more general,
a coefficient,
q et 'v,
ao, = e2
ds
J
-ao f J
(52)
can be defined such that,
P =
d 2 Jjwj
(53)
is the implicit current. The ordering of Eq. (37), in terms of the small parameter,
A/a, follows from the general conditions stated above. In particular, one seeks
solutions where the distribution function, f, evolves on the transport time scale
or at rate of order (A/a)2 slower than the basic collision frequency. Other
parameters are related to A/a to achieve a maximal ordering where as many
processes as possible appear in the transport equations. Thus, the coefficient, V,
is treated as order, A/a, so that the kinematic flow operator is ordered,
-
a
Vf +
'401Vf = O(A/af),
(54)
a
(55)
a'Vf = 0((A/a)2 f).
In this ordering, the power dissipated in the system by the external forces,
V, balances thermal conduction in the energy transport equation. Here the
18
kinematic flow coefficient, defined in Eq. (38), has, for its velocity space components, been decomposed into an ordered sum, a 1 = ao + al , such that the
leading order piece is equal to the purely kinematic coefficient, Eq. (52). This
means that the statistical aspects of the average on Eq. (38) are relatively weak
for, this term.
Similarly, the scattering operator, C(f), is decomposed into the ordered
sum,
C(f) = CO(f)+C
1 (f)+C 2 (f)
a
a
--D.f+
a
a
D'.5-f+-
a
2
D 2-
a
f,
(56)
where Do contains only velocity components,
Do = Di 1 ejej + D12 (ele2 + e2el) + D22 e 2 e 2 ,
(57)
D' contains off-diagonal, radius-velocity components,
D' = D13 (eie3 + e3 ej) + D' 3 (eae 3 + ese 2 )
(58)
and the diagonal radial components, D3 3 , as well as other small contributions to
the other components, appear in D 2 . In addition, Do and D' have the property
that
0 -w=
DO
D1 -w
=0,
(59)
where wl = 8H/8J 1 . This condition, Eq. (59), assures that all scattering
processes which do not conserve the energy of this particular particle species
do not appear until second order and can be accounted for in the transport
equation.
The Fokker-Planck coefficients, D = (AJAJ/At), are Lagrangian functional integrals averaged over the various stochastic processes and cannot, in
general, be separated into a sum of turbulent and collisional parts. Nonetheless,
it may be useful for physical interpretation to assume such as separation can
be made, at least approximately. In that case, Do and D' will include only
19
collisional scattering, whereas D2 will include turbulent scattering of all action
components in addition to collisional radial diffusion, and collisional processes
involving energy exchange like ion-electron temperature equilibration.
Following this ordering of the kinetic equation, we expand f in powers of
A/a. The zero order equation is,
0 = C0 (fo).
(60)
Now, with the property given by Eq. (59), fo can be any function of H and
J3 [provided the J 3 dependence of H is negligible as assumed by condition
Eq. (36)]. This gives an arbitrariness to the solutions that cannot be properly
resolved by the linear scattering operator given by Eq. (56). However, if like
particle collisions were included in the analysis, the local H-theorem of Ref.
12 would apply and uniquely require fo to be a local Maxwellian, fo = fm =
N(J 3 ) exp(-Ho(J1)/T(J3 )). To conform with the more general theory we utilize
this specific equilibrium. For reasons that will be clear shortly, we normalize fm
as follows:
fo = fn = n(21rmT)-3/ 2 exp(-Ho/T),
(61)
where n = n 3 aJ 3 /OV= n32781/aV is the density per unit spatial volume.
Having solved Eq. (60), we proceed to the first order equation
-.Di
J- - [Vao fol -
(62)
5- f 61 = Co (f1)
The second term on the left side of Eq. (62) becomes, using Eq. (59),
aj- ah
a
+
A.D
af
'9
J
- D -e---
-
(
-D
e3
)
(63)
In the external force term, since a0 describes a kinematic flow in a canonical
phase space, 8/8JI - ao = 0, one can write
20
j
-
*-Va* fo=Va
L
-wfO
Using these properties, the first order equation can be written as either
CO(fi)
-a VfO -
Co(fA) =-a
w fo
-D
-e.-'-)
(65)
or
- D - e3
-
8
fo
(66)
Inversion of the collision operator, Co, is now required to determine fi, subject
to certain integrability conditions. In the present formulation these conditions
are trivially satisfied, as we now show.
The integrability conditions are determined by the annihilators of Co, in this
case the particle and energy moments. It is evident that the particle moment,
f d 2 J 1 , also annihilates the source terms from Eq. (65). For the energy moment,
we compute f d 2 J 1 Ho(J), from Eq. (65), and integrate the right side by parts.
There results the condition,
0=-
d2Ja -wfoV+
d2JjwI-D' .e3
(67)
The first term vanishes since the equilibrium distribution, fo, carries no current
in the sense of Eq. (53). Using Eq. (59) the second term is zero.
For use in the ensuing transport equation, it is convenient to express the
driving terms for fi is terms of thermodynamic forces Ai,
A1 = d In n/dJ3
(68)
A 2 = d In T/dJ3
(69
A 3 = V/T
21
(70)
Using Eq. (61) for fo, the first order Eq. (66) becomes Co(fl) = a1 A + a 2A 2 +
c 3 A 3 , where the coefficients ai, are given by
a2 =
(71)
-[D-efo
aJJ-
Cil=-
- [D
--
e(
fo(72)
-
(73)
a3 = -a0L -wfo
, and to define the linearized collision
It is now convenient to write fi = fo
operator, C, by
(74)
COMM = Ct( 1),
where, by virtue of Eq. (59),
C,(f 1 ) =
49
9
- Dofo - 41
,
(75)
and C1 is clearly self-adjoint. Moreover, this factorization of f, remains useful
when binary collisions are included in Co, so that the operator, although more
complicated, retains the self-adjoint form. If the perturbed distribution, fl, is
expressed as a sum over the thermodynamic forces,
g=Ai,
(76))
then the individual responses, gi, are determined from
Cf(g1 ) = ai
To second order in A/a, Eq. (62) becomes,
22
(77)
0 + -a3(J)fOV
at
aJ
h3 '
+
'_
-a_(J)fiV + a
TJi
a' foV - C 1(fl) - C 2 (fo)
(78)
= Co(f 2 )
The integrability conditions for the solution of f2 now provide the transport
equations. Accordingly, although the transport equations themselves are second
order, they require knowledge of f only through first order,
The particle moment of Eq. (76) is,
a
ynf
3
af d2f2Jja fOV
+ 5j
3
-
dd 2 Je
3
a
-D
-
I
d2 JD
33
(79)
=0
The second term in brackets can be integrated by parts,
d2Je
3 -D' ~Jd2aj_±
-
Afi =
d2Jf
a
(e3 -Dl)t
fJd2Jf,0. -D1 -e 3
f
(80)
2Jaifi
using the symmetry of D1 and the definitions Eqs. (71)-(73) to give,
an3
t
a
[[
J3 f
d2TJafOV -fd2j
u
f
f,
a
This is in the form of a transport equation
23
d2JI D 33 9=O 0
(81)
an- + -
at
ah
=0
(82)
with the flux, 1, being proportional to the thermodynamic forces, A2 . Note that
the thermodynamic force, A 1 , is defined in terms of the spatial number density,
n, while the transport equation itself is expressed per unit J3 . One can now
define the transport coefficients, T 2 , as follows,
din n
r = Ti, d
+
din T
12
VT
dJT + T13T
(83)
We use an inner product notation, (g, h) = f d 2Jjgh, to write the coefficients.
The second term in Eq. (81) by virtue of Eqs. (76) and (77), yields implicit
contributions of the form T'i = -(al, g,). In addition, there are explicit
contributions to the flow that do not require the calculation of fl. We denote
these by Ti 3 . Collecting terms, the particle transport coefficients, Ti, become,
Ti = (-D
3 3,
fo) - (ai, g1) = Til + T'l
T1 = (D 33 , fo
-
-
(ai, g2) = T12 + T12
T13= (Ta3 , fo) - (a 1, g3 ) = T13 + T'13
(84)
(85)
(86)
The coefficient, T11, is the particle diffusion coefficient, appropriately normalized,
and contains both explicit and implicit parts.
The procedure for obtaining the energy transport equation from the energy
moment of Eq. (78) is similar. The first order term in the external force, V, now
contributes to give power input. The energy transport equation is in the form
--- njT +
3 rT + q)= IV +Pe
wt 2 hJa f2
with the heat flux,
24
(87)
q/T = T21
dlnn
dJ3
+ T22
dlnn
- + T 32 V/T
dJ3
(88)
and explicit heating power
Pe
d2J HO(J)
-f
al foV
(89)
The transport coefficients are again decomposed into implicit and explicit
parts, as
T21 =
-Dfa fo
-
T22 =
-D33, fo
-
T23 =
Ta3 , fo(H
(as, gi)
-
-
(a2, g2) = T 2 2 + T' 2
(as, g3) = Te3 + T'a
-
-
T21 + T
(90)
(91)
(92)
Finally, the current, P, appearing in the power input term I V of Eq. (87),
can also be expressed as a flux driven by the thermodynamic forces
i=T77dlnn
dnn +T 3 V/T
(93)
where the transport coefficients in Eq. (318) are only of the implicit type,
gi)
(94)
T32 =
-(a3, g2)
(95)
T 33=
-(a
, g3)
(96)
T31
=
-(a3,
3
When the equilibrium Hamiltonian, Ho(J), is known, transport equations Eqs.
(82) and (87), together with the coefficients in Eqs. (84)-(86), (90)-(92), and
25
(94)-(96) are a closed system evolving the density and temperature. The particle
density and current are expressed naturally per unit J3 in this representation.
However, the Hamiltonian, H1O(J), depends implicitly on the magnetic field. This
requires knowledge of the current density in real space and includes the kinematic
contribution, IP, in addition to I'. Were it not for this circumstance the real
space densities would never be needed in the transport theory. The appearance
of n in the thermodynamic force, A,, is a consequence of the normalization, Eq.
(61), and not fundamental.
In addition to the dissipative current, P, there are diamagnetic currents,
Ie, computed above, for which no work is done by the external force. The flux
surface averaged current, Eq. (51), to first order in A/a has an implicit piece,
as given by Eq. (93) and an explicit piece in the form
dlnn
1 , = T31 dj- + T32
b d rdJ
dlnT
3
obtained from the second term of Eq. (51) using f = fo.
26
(97)
WY.
General symmetries of the Transport Matrix
Consider first the matrix, Tij, of transport coefficients derived in Sec. III.
This is decomposed, Tij = T + T! , into an implicit part, T , which requires
the calculation of a first order distribution function, and a part, TIJ, which
is given explicitly in terms of the Maxwellian distribution, fo, and the FokkerPlanck coefficients.
The symmetry of the implicit coefficients follows immediately from the selfadjointness of C,
T' = -(ai, gj) = -(Ct(gi), gj) = -(gi, Ct(g3 )) = -(gi, aj) = T
(98)
Equality of the explicit coefficients, T' 2 and T j,1 is evident by inspection of Eqs.
(84)-(86) and (90)-(92). This leaves the explicit coefficients T'1, T', and T23 , T 2
without any general symmetry relation. Noting that these coefficients represent
kinematic flows, we can define a kinematic transport matrix, Tk, consisting of
these coefficients with all other elements zero. The rest of the matrix, T - Tk,
will be designated, T. To summarize,
T=Ts+T
(99)
k
where the part due to scattering processes
T11
T1 2
T= T21 T22
T.
T2
T13
T23
(100)
T733
is symmetric, and the part due to kinematic flows,
Tk =
0
0
0
0
T11
iT i
T l
0
is, in general, not symmetric.
27
Te
(101)
The symmetry of the matrix T can be traced back directly to microscopic
reversibility. Mathematically, this gives rise to symmetric Fokker-Planck equation as demonstrated in Sec. II, which accounts for both the self-adjoint property
of Ct, and the equality of the forcing functions, ai, for fl, in Eq. (77), with the
moments, ai, defining the fluxes in Eqs. (82) and (87). Physically, the fluxes at
conjugate locations in the matrix T' correspond to scattering processes that are
microscopic inverses of one another.
On the other hand, the fluxes at conjugate locations in the matrix, T ,
correspond to independent kinematic flows. The coefficient T' 3 describes a radial
flow driven directly by the force, VIT, while T', comes from a current within the
flux surface due to finite orbital widths and the density gradient. Such processes
cannot be linked by time reversal and are therefore not conjugate in the Onsager
sense. This is why the matrix, Tk, does not possess a general symmetry.
It is enlightening to compare these symmetries to those arising in a different
type of transport theory that corresponds more closely to the empirical theory
of Onsager. The comparison is considered in the following section.
28
V.
Onsager's Transport Theory and Restricted Symmetries
Onsager's transport theory 2 is based on the regression or relaxation of
fluctuations about a global thermal equilibrium. The transport equations are
not directly derived but result from an empirical construction (described below)
that relates the relaxation of the global moments to the fluxes. The fluctuations
can be described by a Fokker-Planck equation where the independent variables
correspond to the global parameters. The interpretation is then quite different
from the Fokker-Planck equation on the single particle phase space employed in
Sec. III above. The Fokker-Planck equation symmetries remain, nevertheless,
and lead to the usual Onsager relations, when this type of transport theory is
appropriate. In the discussion below we will attempt to compare and contrast
the two transport theories, particularly as to their regimes of validity. The
principal difference is the requirement, in the Onsager type theory, of a nearly
global thermal equilibrium. The theory of Sec. III, in contrast, requires only
nearness to a local equilibrium and is not in many cases of practical interest
close to a global steady state.
We now give a specific, but readily generalizable, example of Onsager's
formulation of transport theory. Consider a medium bounded at z =+a, by
perfectly insulating walls and homogeneous in the y and z directions. The
system is characterized by spatial distributions of energy E(x), density n(x),
and a variety of other possible quantities, Q'(x) such as magnetization, charge
etc. In equilibrium, these distributions are, on average, constant functions of z,
but undergo fluctuations about this constant value, as illustrated in Fig. 1. The
spatial distribution of energy can be characterized by the global moment,
dzzE(z),
al =
(102)
and similar moments characterizing the other quantities,
dxxQ'(x).
a=
(103)
If one averages statistically over all distributions, when the system is in equi-
29
I
librium, the average value of these moments is zero, & = 0.
Now one can take a subset of the equilibrium ensemble (equivalent to a nonequilibrium state) where for each i , all the ai's are non-zero and identical. For
this non-equilibrium ensemble, each distribution, Qi(x), has an average gradient,
dQil/dz, which is non-zero. For small departures from equilibrium, a' small in
some sense, this average gradient should be proportional to the moment, ai, thus
(104)
Ca,
where Qi is the equilibrium value, and c is a constant. The system will now
relax to equilibrium as the moments at approach zero with rate of transport, or
flux Ji given by
J=
(105)
dt
Again, this average is taken over a non-equilibrium ensemble. Equations (104),
(105) are the relations necessary to relate the regression of fluctuations to transport theory. Now, if there is a dynamical theory expressing the relaxation of the
moments, one can write,
(106)
.7
This is clearly a global dynamical theory in which the coefficients depend
only on the equilibrium values, Q, and are independent of x. To construct a
transport theory one first uses Eq. (106) to rewrite the flux as,
J =.-.T
C
j
T
dx
(107)
Then the isolated system of Fig. 1 is embeded in a larger system, allowing slow
spatial variations of the equilibrium values, Qi --+ Qi(x). By assuming that Eq.
(107) is still valid for this slice of the larger system then, using the conservation
equation,
30
+
at
ax
= 0,
(108)
together with Eq. (107), one can obtain local transport equations,
.
at
-T'a -Qj = 0
3x
(109)
This, in essence is Onsager's argument for obtaining the transport coefficients,
T', from the regression coefficients, Tj, of Eq. (106).
The linear regression theory for the fluctuations can be obtained from
the Fokker-Planck equation, where the independent variables are the global
moments, a3 ,
ft
-
a
a
:H-1f +
18
a
-- JD - -2
a&i J
(110)
The equilibrium distribution, fo, in the absence of kinematic flows, H' = 0, is
given by, fo = J, the phase volume. For the system considered here, we expect
J to be Gaussian in the fluctuations, a3 , for (macroscopically) small values of
ck',
J = Aexp(-gjaa'),
(111)
where A is a normalization constant, and gi1 is a symmetric matrix. Equation (111) is the form expected from statistical mechanics where the exponent,
-gija' ae, represents the entropy and takes its maximum value for the equilibrium position, a' = 0. By the same considerations, we expect the diffusion
tensor, Dj, to be independent of the a', for the values of a' considered. Except in certain special cases9 the kinematic flow term precludes an equilibrium
solution of Eq. (110) and would have to be excluded from a regression theory.
Here we include it as a small additive and non-symmetric contribution to the
flux derived below.
f,
We now assume the system to be perturbed to some non-equilibrium state
where the ai's have non-vanishing mean values,
31
(112)
J {da}aif
&
The regression of these mean values, a-?, is obtained from the first moment of
Eq. (110),
+ 2
f{d}Hf
49t
=
{da}Hif -
J
-- In J
9ai
{da}fD
{da}fDk
9gki C
(113)
Defining the kinematic flow,
f{da}Hf ,
fj
(114)
and the regression coefficients,
T-i =- D
i ,(115)
we have,
--x.
-
=
dt
ai
(116)
The matrix Tj is clearly symmetric , since both Dik and gki are symmetric. Note
that it is necessary even here to extract the kinematic flows before symmetric
regression equations can be obtained. This fact has been noted previously by
several authors- 9 .
32
VI.
Conclusion
The general theory of the Fokker-Planck equation and its symmetries has
been reviewed. The equation possesses a partial group of symmetries resulting
from microscopic reversibility for processes obeying detailed balance. These
symmetries in the kinetic theory hold whether or not the more restrictive conditions required to derive a transport theory are obeyed. The transport theory
symmetries depend critically on specific system properties.
Onsager considered systems near a global thermal equilibrium state and had
a Fokker-Planck equation describing the evolution of the distribution function of
global variables. The assumption of a global equilibrium eliminates certain nonstationary, kinematic and radiative processes. A symmetric regression theory
for the relaxation of these global parameters can then be derived. By regarding
the isolated system as a slice embedded in a larger system, one gets a transport
theory obeying the Onsager reciprocal relations.
A more general transport theory, derived in the present paper, uses a
Fokker-Planck equation for a distribution function on a single particle phase
space. Here, the developement of a transport theory requires only closeness to
local thermal equilibrium and the transport equations are directly derived. Nonstationary kinematic and radiative processes are allowed. The transport theory
has broken symmetry in general although a symmetric portion corresponding to
Onsager's reciprocity can be identified.
REFERENCES
1.
Lord Kelvin (Sir W. Thomson), Collected PapersI, pp. 232-291, Cambridge,
1882.
2.
Onsager, L., Phys. Rev. 37, 405 (1931); 38, 2265 (1931).
3.
Casimir, H.B.G., Rev. Mod. Phys. 17, 343 (1945).
4.
Ware, A.A., Phys. Rev. Lett. 25, 916 (1970).
33
5.
Enskog, D., Inaugural Dissertation, Uppsala (1917); also, Chapman, S.
and Cowling, T.G., The Mathematical Theory of Non-Uniform Gases,
Cambridge (1970).
6.
7.
Van Kampen, N.G., Physica 23, 816 (1957) and Physica 23, 767 (1957).
Graham, R., and-Haken, H, Z. Physik 243, 289 (1971).
8.
Zwanzig, R., Phys. Rev. 124, 983 (1961).
9.
Mori, H., Prog of Theoretical Physics 33, 423 (1965).
Molvig, K., Lidsky, L. M., Hizanidis, K., and Bernstein, I. B., Comments
on Mod. Phys.:Part E, 7, 1134 (1982).
10.
11.
12.
Rosenbluth, M. N., Hazeltine, R. D., and Hinton, F. L., Phys. Fluids 15,
116 (1972); Hinton, F. L. and Hazeltine, R. D., Rev. Mod. Phys. 48, 239
(1976); Bernstein, I. B., Phys. Fluids 17, 547 (1974).
Bernstein, LB. and Molvig, K., Phys. Fluids 26, 1488 (1983); Cohen, R.
H., Hizanidis, K., Molvig, K., and Bernstein, I. B., Phys. Fluids 27, 377
(1984).
34
APPENDIX: Microscopic Reversibility and Detailed Balance Conditions
A turbulent plasma is characterized by the occurrence of enhanced fluctuations
due to some unstable collective modes the plasma can support. These enhanced
fluctuations always affect to some degree the rates of the various relaxation
processes. A growing in amplitude electromagnetic perturbation, for instance,
can easily change or even destroy the structure of the phase space orbits which
corresponds to the integrable part of the problem. If the latter has a Hamiltonian
denoted by H 0 , then the overall Hamiltonian can be modeled as,
H = Ho +H
(Al)
1
where is a small parameter of statistical nature. The basis of this model is
the sensitivity of the unperturbed orbits to the initial conditions; a growing
electromagnetic perturbation can turn them completely irregular. In low level
turbulence cases, Hj can easily be considered as a perturbation to the Hamiltonian Ho. One usually includes in Hj scattering processes due to particle-toparticle interactions (Coulomb collisions). Therefore H can be modeled to be
responsible for all the relaxation processes occurring in the system as they are
being modified by a low level turbulence and collisionality.
The dynamical equations of motion in canonical form are
Y=FC(Y, t; )=J -
(9H
(A2a)
(A7y
satisfying the initial conditions
Y(t = 0) =Yo
(A2b)
where Y corresponds to a phase space point and J the symplectic matrix.
Although the parameter
is going to be modeled as a statistical quantity, it
expresses the various couplings occurring in the system in a deterministic way.
Equation (A2) does not need to be integrable of course; however, we assume
the existence and uniqueness of the solution. The formal solution in generalized
coordinates (not necessarily canonical) X as well as in the original canonical
coordinates Y is
# 357
= U(t; )Xo
(A3a)
Y = V(t; )Yo
(A3b)
X
with
X= OY
(A3c)
where U(t; ) (V(t; )) is the operator which propagates the system from the point
Xo (Yo) to the point X (Y) during the time interval t taking into account all the
various interactions (C); 0 is the transformation operator X -+ Y. The relation
between the operators U(t; ) and V(t; C) is,
U(t;. ) =0'~;
(A4)
where 0 -' is the inverse operator of 0. One can now introduce the time reversal
operator T which has the following properties
Tt = -t
T g(X 1,tl;X 2 ,t 2 ;-
(A5a)
;Xntf) = g(TX,-tn;- ; TX 2,-t 2 ; TX 1,-tl) (A5b)
T 2 =r
(A5c)
- - - tn; as a concequence, one
t2
where I is the identity operator and t1
has for the Jacobian determinant J(X -+ TX) of the mapping X -+ TX
J(X -+ TX) = 1
(A5d)
Invariance of Hamilton's equations under time reversal implies that the principle
of microscopic reversibility can be expressed as follows,
36
T V(t;)=
(A6)
V-(t;
where V-1 denotes the inverse operator of V and C. Equations (A4-A6) then
imply
W'(t;
-) C) = U(-t; TC)
TLU(t;
(A7)
where U-1 denotes the inverse operator of U. Equation (A2) describes an orbit
in phase space which originates from the point X 0 .
Now consider an ensemble of initial points distributed according to a phase
space density
N(X, t = 0) = 6( 6)(X -
X0 )
(A8)
As all points move according to Eq. (A2) the density function is being conserved
since no orbits are being added or eliminated. Therefore, one has,
aN
8
Ft +
-. [F,c(X, t; )N] = 0
(A9)
where the generalized force F,, is related to the force Fc of the canonical
description through
F
=
ax
- Fe
(AlO)
The formal solution of Eq. (A9)is,
N(X, t; , Xo) = J(X)N(U-'(t; )X, t = 0)
(All)
The factor J(X) is the Jacobian determinant det(aU~1 X/OX) of the mapping
Xo -- X.
We are now in position to model the perturbation as a stochastic process
with being distributed according to the probability density function P( ). The
joint probability, f 2 (X, t; X', t') is defined as follows,
37
f2 (X,4; X', t')
(A12)
f {dXo}Po(Xo) fdCP(C)N(X, t; C)N(X'; t'; ()
where the probability PO(Xo) refers to an ensemble of initial conditions. Using
Eqs. (A8) and (A12) one has,
f 2 (X, t; X't') =
{dXo}Po(Xo)
dCP(C)J(X)6[U-1 (t; )X - Xo]
(A13)
J(X')6[U1(t'; C)X' - XO]
Upon using the transformation property of the delta functions: 6[U- 1 (t; C)X - Xo]
6{U-1(t; C)[X - U(t; C)Xoj}= det(8'U-X/OX)6[X - U(t; )Xoj =6[X' - U(t'; )Xo] J- (X)
Eq. (A13) becomes
d P(C)
{dXo}Po(Xo)
f 2 (X, t; X', t') =
5[X - U(t; )Xo]1[X' - U(t'; )Xo]
(A14)
Under time reversal the above equation takes the form (Eq. (A5))
f 2 (TX',
-t;
t) =
J
{dXo}Po(Xo)
J
d P(e)
6[TX' - U(-t'; )Xo]b[TX - U(-t; )Xo]
(A15)
Using Eq. (A7) for microscopic reversibility in Eq. (A15) yields
f 2 (TX', -t'; TX, -t) =
{dXo}Po(Xo)
6[TX' - U~ (t'; T1
38
J
d P( )
)Xo]b [TX - U~-(t; T
)Xo](A16)
which by virtue of Eqs. (A5) and (A7) becomes
f 2 (TX', -t'; TX, -t) = J{dX}P(Xo)
J
dCP(C)
[X' - U(t'; T- 1 C)T- 1 Xo]6[X - U(t; T'--)T~'Xo],(A17)
or, since dCP(C){dXO}P(Xo)=dT- 1 CP(T -' ){dT -'Xo}Po(7-'Xo), upon changing of integration variables in Eq. (A17) one obtains
fh(TX', -t'; TX, -t) =
{dXo}P(Xo)
dCP( )
6[X' - U(t'; )XO]6[X - U(t; .)Xo]
(A18)
Comparing Eqs. (A18) and (A14) gives the symmetry relation for the joint
probability under time reversal,
f 2(X, t; X', t') = f 2(TX', -t'; TX, -t)
(A19)
For processes with external influences (time independent on the time scale of
interest) this condition can be written as follows
f 2 (X, t; X', t';{X}) = f 2 (TX', -t';
TX, -t; {TX})
(A20)
where {X} corresponds to a set of externally controlled parameters or constraints
(magnetic field in particular for systems of interest).
For stationary Markov processes the joint probabilities are simply related
to the conditional probability density P(X, tjX', t') ,which is called transition
probability and depends only on t - t', and the one point stationary distribution
function f(X') :
f 2(X, t; X', t') = P(X, tjX', t')f(X')
39
(A21)
Then Eq. (A19) yields for the transition probability
P(X, tjX', t')f(X') = P(TX', -t'j TX, -t)f(TX)
(A22)
Takng now into account the relation between the volume elements {dX} and
{dX'}
J(X)
J(X')
_
{dX'}
{dX}
(X)A23)
f(X')
and since J(TX) = J(X) by virtue of Eq. (A5), one finally has
P(X, tAX', t')J(X') = P(TX', -t'ITX, -t)J(X)
(A24)
Equation (A24) expresses the detailed balance condition for stationary Markov
processes.
In general, for non-stationary processes, one can formally add a piece, Pn.,
in the right hand side of Eq. (A24) which accounts for all those proccesses
associated with non-stationary behavior (accelerating particles, for example)
P(X, t|X', t')J(X') = P(TX', -t'I TX, -t)J(X)
+ P.,(X, t;X', t')
(A25)
Time reversing the above equation yields
P(TX, -tTX', -t)J(X') = P(X', t'lX, t)J(X)
+ TPn,(X, t;X, t)
(A26)
where the time reversed P,,.(X, t; X, t) is to be taken according to the rule
expressed by Eq. (A5b). Interchanging (X, t) -+ (X', t') in Eq. (A25) yields
P(X', t'IX, t)J(X) = P(TX, -tj TX', -t')J(X')
+ P.,(X'; t',X, t)
40
(A27)
Comparing now the last two equations yields the following property for P,,.
TP.,(X, t; X', t') = -P,(X', t'; X, t)
41
(A28)
FIGURE CAPTION
f.
Fluctuations about equilibrium in an isolated system. The moment al =
dxxE(x) describes the asymmetry of the distribution for a specific fluctuation.
42
al/Eo
E W)
-
-
-
-
-
~~*1'
_____________________________________________________
x= -a
~
X:O
1
x~a