Diffusion processes in nonequilibrium thermodynamics

DIEGO BRICIO HERNANDEZ CIMAT

Apartado Postal 402 36000 Guanajuato, Gto.

MEXICO

DIFFUSION PROCESSES

IN NONEQUILIBRIUM THERMODYNAMICS (*)

(Con ferenza t enu ta il 18 marzo 1988)

ABSTRACT. - - This paper deals with two alternative formulations in irreversible thermodynamics: a microscopic, stochastic description is associated with every deterministic, macroscopic model. The association is done via semigroup theory, and is used in order to investigate basic issues such as equilibrium states and the stability thereof. In particular, the density of an invariant measure is characterized as an integrat ing factor for the infinitesimal generator of the process.

The stochastic model is a diffusion process in configuration space, and it can be reconciled with dynamics via Smoluchowski's approximation. An ergodic property holds, and the probabilistic representations of solutions to both initial and boundary value problems provide the averaging techniques l inking macroscopic observables to microscopic quantities. Thus, a new physical interpretation of the stochastic calculus of diffusion processes is given, and with it a natura l dual formulation of irreversible processes.

1. - INTRODUCTION.

Natural processes are irreversible, i.e. they occur in a preferred direction of the time axis. The Second Law of Thermodynamics states that natural processes evolve towards an equilibrium state, and gives entropy increase as a quantitative measure of that tendency.

This paper deals with irreversible thermodynamic phenomena from the point of view of stochastic processes, very much in the

(*) This talk was given at the Seminario Matematico e Fisico di Milano (Politecnico e UniversitY) while the author was associated with the Centro di Teoria dei Sistemi/CNR, under the patronage of the Consiglio Nazionale delle Ricerche (Italy). The invitation of Prof. G. Prouse is gratefully acknowledged.

28 D . B . HERNAND~Z

spirit of [Propp, 1985]. However, the concepts presented here are independent from those in Propp's thesis and, in the author's opinion, provide new physical insight into the role played by stochastic processes in thermodynamics. We have striven to keep the treatment as simple as possible throughout, t rying to render the concepts accessible to both physicists and mathematicians.

In the following sections we study thermal processes taking place in a bounded portion B of physical space (B open in ~ , with d _< 3), assuming the system in question is closed i.e. that it exchan- ges neither matter nor energy with its surroundings. The usual assumptions of local equilibrium and linearity (i.e. Fourier's law) are introduced, thereby resulting in a mathematical model for temperature's space-time behaviour which consists of parabolic partial differential equations valid in (0, ~ ) X B, duly supplemented with Neumann conditions specified on the boundary aB.

Such boundary value problems (BVP's) are characterized by means of linear second order elliptic operators defined for smooth u : B . - - > ~ by

1 (la) A u : - - 2 (a, V 2 u ) + (b, V u ) + c u ,

where a is d X d-symmetric matrix valued, b is d-vector valued and c is scalar, all of them continuous functions on B. In (la), the scalar product ( . , . ) is defined by

(lb) (A, B) : ~ tr(ABr), for A, B E ~ x n .

To ensure that A is sufficiently well behaved, it is convenient to re- strict ourselves to the uniformly elliptic case, in which the eigenvalues of the symmetric matrix a(x) lie in a fixed interval [~,A, with 0 < ~ < ~ < ~- ~, for each x ~ B. Assuming 0B is smooth, there is a continuously turning unit normal n(x) for each x E aB, hence the relevant boundary operator is of the form

(lc) Bu(x) : ------ (n(x), a(x) Vu(x)), x E OB.

Thus, the BVP's of interest are of the general form

(2) ut~--Au--O in (0, ~) X B; B u ( t , . ) - - O in aB, for t >_ 0,

where r : B --> ~ is continuous and represents an internal produc- tion term. The main problem to be posed on (2) is an initial value

DIFFUSION PROCESSES IN NONEQUILIBRIUM THERMODYNAMICS 29

problem (IVP), which calls for a solution to (2) satisfying a given

initial condition u ( 0 , . ) - - ~ . One such initial condition constitutes a steady state for the dynamical system (2) if the solution of the corresponding IVP is time independent i.e. if the following elliptic BVP is satisfied:

(3) A ~ - - - r in B; B ~ - - 0 in 3B.

A constant steady state is termed an equilibrium state. A solution to the S V P (2) is called a transient regime if it decays towards an equilibrium state with the passage of time.

Section 2 below presents a number of elementary examples il- lustrating this circle of ideas, all of them involving thermodynamic processes in one space dimension, without internal energy produc- tion (r and with c ~ 0. It is shown in each example that a transient regime does originate in any initial state away from equilibrium, so that deviations from equilibrium always die out as t--> (Propositions 1 and 2). The fact that the system is closed (i.e. the Neumann boundary conditions) is seen to play a most prominent role in all this, as it ensures the existence of a zero eigenvalue of A, from which the equilibrium state the transient decays to is obtained, together with an estimate of the rate of approach to equilibrium.

Later on, section 3 presents the construction of a microscopic alternative to the macroscopic models of thermodynamic phenomena embodied in BVP's like (2) and (3). Such a microscopic model is random in character, and consists of a diffusion process whose

trajectories stay within B, bouncing back into the interior of the set once they hit 3B (the boundary is a reflecting barrier for the diffusion). This is achieved via the submartingale problem construction [Stroock-Varadhan, 1971], the diffusion process corresponding to a given elliptic operator A being that one whose generator is A - c supplemented with the Neumann boundary conditions already referred to. Alternatively, section 4 contains a direct construction of the diffusion corresponding to A - c in the case d ~ 1, assuming brownian motion with reflecting barriers at 3B has been constructed by independent means (Proposition 6); this direct construction ori- ginates in work reported in [Csink-0ksendal, 1983].

In addition, Semigroup Theory offers a natural way to define equilibrium and stability thereof for the alternative stochastic model �9

30 D.B . H~NANDE~

in equilibrium, microscopiv behaviour must be described by an invariant probability measure. The existence of an invariant measure is guaranteed by the fact that the state space of the diffusion is compact [Kunita, 1971] ; in addition, the treatment given below cha- racterizes an invariant measure as a suitably normalized integrating factor for A.

For a closed thermodynamic system, equilibrium properties are shown to be given as averages with respect to the invariant measures e.g. (6.14). Away from equilibrium, macroscopic variables can be recovered from their microscopic r dynamica l , counterparts by forming suitable averages, as specified by the probabilistic representations of the solutions to the IVP associated with (2); see (3.7). Moreover, there is ergodicity under the assumption that the transition probabilities have densities (Proposition 3). Thus, a Statistical Mechanics has been defined.

This Statistical Mechanics is not quite satisfactory from the physical point of view, however, since no dynamical quantities such as mass and acceleration figure in it. Even so, this construction may represent a convenient means for calculating macroscopic properties using Monte Carlo techniques. To let it offer more than just a convenient computational device, section 5 studies the reduction of the dynamical Ornstein-Uhlenbeck diffusion model to the diffusions obtained as described above, which thereby appear as the Smoluchowski approximations to the corresponding Ornstein-Uhlen- beck model. This is done in the case of a constant coefficient matrix a, building upon Theorem 10.1 in [Nelson, 1967]. That this could be done was initially suggested to the author by M. Pavon (Padua), to whom recognition is extended.

Finally, section 6 offers some kind of r summing up ~> of all the above.

2. - SOME STABLE EQUILIBRIA.

Consider a slender, uniform metal bar of fixed length, thermally insulated from its surroundings and subject to a nonuniform temperature distribution (or profile) along its length. A well known and widely verified empirical law, applicable to such kind of nonequilibrium situation is Le Chatelier's Principle, according to which

. . . i f equilibrium is perturbed, the system's parameters will evolve in such a way that equilibrium is restored ~. Equilibrium, of course,

DIFFUSION PROCESSES IN NONEQUILIBRIUM TH]~MODYNAMICS 31

means thermodynamic equilibrium throughout the bar, i.e. a uni form temperature distribution.

Clearly, this empirical principle is but a part icular instance of the Second Law of Thermodynamics [Callen, 1960], and it merely asserts the asymptotic stability of thermodynamic equilibrium states. Moreover, it can be readily obtained as a theorem in the mathematical theory of heat conduction. Indeed, let the bar's length be 1, when measured in convenient units, and let a be its thermal diffusivity, assumed to be constant. Let u t ( x ) - u(t, x) (t >__ O, x E [0, 1]) be the temperature at point x, at time t, and assume u( . , x) is C ~ and u(t, .) is C ~ for each fixed x and each fixed t, respectively. Then, assuming Fourier 's law of heat conduction [Bird et al., 1961], the law of conservation of energy translates into

0u i~u (la) 0t - - a - ~ - ' in (0, oo) X (0, 1),

au au (lb) 0-~ (t, O) ~-- O, ~-~ (t, 1) = O, t ~ O.

An easy application of the method of separation of variables to this boundary value problem (BVP) leads to the eigenvalue problem

y" ~ 2y in (0, 1); y'(0) ~ 0, y'(1) --- 0 ,

whose spectrum is - - k ~ n~(k ~ 0, 1, 2, ...) and whose eigenfunctions are cos k~x(k ~--0, 1, 2, ...). Given the initial condition u(0, x ) ~ f(x), with f E C~[0, 1], it suffices to chose {C~} in such a way that

f(x) ~ ~ C~ cos k~nx k ~ 0

to obtain the solution to (1) in the form

oo

u(t, x )~ (f) + ~ C~ e-ae~'tcos k~x, k = l

where the average {f) is given by 1

(2) (I) f f(x)d 0

Therefore, we have

32 D . B . HERNANDI~

PROPOSITION I.

In a thermally insulated metallic bar, for every initial temperature profile f there is a unique transient regime u starting from f. Moreover, u (t, x) --> ( f ), t--> ~, with (f) as in (2).

The first part of this result is valid even if the Neumann BC's (lb) are replaced by Dirichlet or mixed type BC's; however, the second part is intimately related to the BVP as it is, since it embodies the thermally insulated character of this system. For a closed homogeneous system, equilibrium is attained when the energy present in the system distributes uniformly over the whole.

Convergence to thermodynamic equilibrium attains even under quite relaxed assumptions concerning the physics of the situation, provided the system is closed. For instance, if the thermal diffusivity a is not constant (i.e. if the bar is not homogeneous from the point of view of heat conduction) then the BVP (1) must be replaced by

Ou O [ Ou] , on (0, ~ ) • ( 0 , 1 ) ,

$u Ou (3b) 0~- (t, O) ~-- O, ~ (t, 1) .----- O, t _> O.

The eigenvalue problem to be solved is then

(4) d [ a(x) dY ] d~ ~ ~ ly on (0,1); y ' ( 0 ) ~ 0 , y ' ( 1 ) = l ,

which is readilly seen to have 0 as an eigenvalue, with any nonzero constant as corresponding eigenfunction. Moreover, f rom the classical Sturm-Liouville theory [Stakgold, 1979] it is known that the spectrum of (4) constitutes a sequence

(5) 0 ~ s > ,~1 > ..., ~ --> ~ ,

with a corresponding sequence of eigenfunctions {~.}, orthonormal with respect to the inner product

(6)

1

(0, ~) : ~ l ~o(x) p(x) dx . u


This is so under the additional assumption that, for some ao > 0,

(7) a E C1(0, 1) and ~(x) _> ao > 0 for all x ,

i.e. in the regular case. By separation of variables, the solution of (3) satisfying the initial condition u(O,.)---f E C1[0, 1] is readily seen to be of the form (f ) + 0(1) as t--> 0% hence Proposition 1 continues to be valid when applied to this BVP. In other words, both the quantitative Le Chatelier's Principle as well as the quantitative characterization of thermal equilibrium are seen to be valid in this more general situation.

Similar considerations apply to systems in which there is both heat conduction and fluid flow, say as in a pipe subject to a nonuniform temperature distribution at a certain instant (t ~ 0), and supposed to be otherwise thermally insulated. In this case, energy conservation requires that u satisfies the BVP

(8a) Ou au O[ Ou ] O-t- + v(x) Ox - - Ox ~(x)

, in (0, ~ ) X (0, 1) ,

au ax (8b) 0-~ (t, O),= O, 0-~ (t, 1) = O, t > O,

where v ~ C[0, 1] is the longitudinal velocity profile. In this problem, the BC's merely express the continuity of the temperature functions at the entrance and exit points.

Given an initial temperature profile, its corresponding trajec- tory can be generated by solving (8), say by separation of variables. This leads to a non self-adjoint eigenvalue problem which, in a standard fashion [Stakgold, 1979, p. 101], can be converted into the self-adjoint Sturm-LiouviUe problem

d Y ] ~ f w ( x ) y on (0,1): y'(O)~O, y'(1)~--0, d w(x) a(x) dx

which always has 0 as an eigenvalue, with a constant eigenfunction. This transformation is achieved by multiplying both sides of the original differential equation by the strictly positive r integrating factor

S e m i n a r i o M a t e m a t i e o e Fisie.o - 3

34 D . B . HERNANDE'Z

Therefore, the spectrum will again have the form (5) assuming again that (7) holds, but the corresponding sequence of eigenfunctions {~,} will now be orthonormal with respect to a new inner product, given by

1

(9a) (~' P) : ~ l ~(x) p(x) e(x) dx , 0

with

(9b) w(x)

~ ( x ) : - -

f w(D d~ 0

instead of (6). An easy computation shows that the solution of the BVP (8) [u (O , . )= f ] is again of the form

(~o) u( t , x ) - -~ ( f ) + O(1) as t - - ~ ,

where

( i f ) <f) : : (f, I ) ,

with ( . , . ) as in (9), insteead of (2). The following analogue of Pro- position 1 holds:

P R O P O S I T I O N 2 .

A fluid, with strictly positive thermal diffusivity, flows through a hollow tube, under a stationary velocity profile and in such a way that the temperature profile is always continuous. Then, for each initial temperature profile f there is unique transient regime u, solution to (8), with f as initial condition. Moreover, u(t, x)--> (f), t--~ ~, with (f) as in (11).

Thus, Le Chatelier's Principle holds for closed thermodynamic systems. Proposition 2 shows that it also holds for some open systems, at least in dimension one, but not necessarily in higher dimension. Thermodynamic equilibrium states are obtained by forming suitable overages of the initial states away from equilibrium, as in (11).


3. - EQUILIBRIUM AND INVARIANCE OF MEASURES.

Clearly, each of the examples given in the foregoing section guarantees the existence of a unique equilibrium solution of a certain linear PDE, plus the global asymptotic stability of such a solution. Let us attempt to give a stochastic interpretation of this situation.

Note that the identically zero function is always an equilibrium solution to the homogeneous BVP's we considered in section 2, and that a nonzero equilibrium solution exists only in those cases when 0 is an eigenvalue. In what follows, it will be assumed that such is the case.

In general, the solution (t, x) ~ u~(z) to the IVP obtained from (2) by adjoining the initial condition u o ~ f can be expressed in the form

e ~kt u,(x) ~ ~ C~ ~(x), k~O

where

and therefore

( ~ , ~z) ~ $~z , k , l ---- O, 1, 2 , . . .

C~ ---~ (f, ~k), b ~ 0, 1, 2 ....

In what follows, let X stand for the interval [0, 1]. Assuming interchangeability of sum and integral, the expression for ut(x) can be rewrit ten as

(la) ut(z) ~ f f (y) Pt(x, dy) , X

where each Borel m e a s u r e Pt(x,.) is given by

(lb)

c o

~k t 1

B k-'~- 0

As to the thermodynamic (i.e. constant) equilibrium solution ~, it is given by ( f ) as in (2.11), which can be rewritten as

(2a) u~-- f f (y) P(dy) , X

36 D . B . HERNANDE~

with the Borel measure P given by

(2b) P(B) : ~- f ~(~) d2 B

The validity of Le Chatelier's Principle (i.e. the global asymp-

totic stability of u-) amounts to the condition that

(3a) f f(y)P,(x, dy)....> f f(y)P(dy) as ~...->~, VfEC(X), x x

i.e. to the weak convergence

(3b) Pt(x,.)--"->P as t - ->~

In addition, the equilibrium character of u means that P is invariant under the flow {Tt} in C(X) generated by the differential equation (2.8a). Such flow is given by

(4) Tt f : --- ~ ,

with ut as in (la). In general the equilibrium character of a temperature profile f for (2.8) an be expressed as

(5) T~f ~ f , t >_ 0

On the other hand, it follows from the Maximum Principles for elliptic and parabolic operators [Protter-Weinberger, 1967] that both P and Pt(x, .) are probability measures, for each $ _ 0. Mo- reover, by uniqueness of solutions to the IVP associated with the BVP (2.8) [Friedman, 1964] one gets the semigroup property of { Tt}, hence the Chapman-Kolmogorov identity

P,+,(x, B) ---~ f P,(z, dy) Pt(Y, B), B

valid for all s, t _> 0, for all z EX, and for all Borel sets B c X. As is well known [Doob, 1953], these conditions imply the existence of a probability space (~, A, Q), as well as of a Markov process { xt , t >_ 0 } with X as state space and such that

(6) Q(xt E Bloco) : P , ( x o , B ) ,

m see (9) below.

DIFFUSION FROCESSES IN NONEQUILIBRIUM THERMODYNAMICS 3 7

Let P~ stand for Q ( . i x ~ x ) , and let E~ denote expectation with respect to P~. Then, the solution to (1.8) satisfying the initial condition uo ~ f is given by

(7) u~(x)----- E~[f(x,)],

thus establishing a bridge between the deterministic and the stochastic formulations.

Process { xt, t _> 0 } can indeed be constructed by a variety of methods, for instance applying the submartingale problem method [Stroock-Varadhan, 1971] to this particular situation. For, let A stand for the closure of the differential operator with domain D (A), defined as

d [ a(x) dU ] du (8) dx -dx ~ v(x)-~x ' x E X

on C2(0, 1)N{uE C1[0, 1]: u'(0)---~-u'(1)~ 0}.

Let 9 denote the space of all continuous functions w: [0, ~)-->X, with the topology of uniform convergence on compacta, and let A be the corresponding family of Borel sets. For each t _> 0, let xt : f2--> X stand for the corresponding projection ~o ~->co(t), a Borel measurable function.

Given u E D(A), x E X and t ___ 0 let

t

Mt(u) : ~-- u(xt) ~ u(x) - - f Au(x~) ds . 0

Following [Stroock-Varadhan, 1971], for each x E X there is a unique probability P~ on (~, A) which makes {Mr(u), t _ 0} into a submartingale for each uED(A) , besides satisfying P ~ ( x o ~ x ) ~ l . The family {P~, x E X} /s said to solve the submartingale problem corresponding to A. The stochastic process {xt} is strong Markov and has {Pt(-,.), t >_ 0} as transition probabilities, i.e.

(9) P~(x~+sEBIx,.,r <_ "D-----P~(x, EB)~---P~(x,,P) P~- -a . s .

for any stopping time r.

This construction provides a stochastic solution to the IVP associated with the BVP (2.8), namely (7).

38 D.B. HEENAND ~z

Moreover, it is a Feller process, in the sense that each T, f 6 C(X) if f 6 C ( X ) i.e. { T , , t _> 0} does indeed define a dynamical system in C(X). The flow {Tt} in C(X) defined by (2.8) in (4) induces a flow {Ut, t >_ 0} in the dual space M:-----M(X,r consisting of all Borel probability measures on (X, r Such a flow can be defined by duality, i.e. in such a way that

(I0) ( T t f , # ) - - ( f , Ut#), t >__ 0

where (~0, #) : = .(q~dp [~ 6 C(X), # 6 M]. x

Fubini's Theorem shows that

An easy application of

(11)

hence Uo ~ ---/~.

u, ~,(B) - - f ~,(az) P,(x, B) , ag

In terms of the stochastic process {z,, t >_ 0 }, Ut # is the distribution of xt, for each t if # is that of zo. Thus, the invariant character of P can be conveniently expressed as

(12) U t P ~ P , t >_ 0

which is seen to be equivalent to (5) in virtue of (10). Correspon- dingly, the asymptotic stability of the equilibrium solution of (2.8) translates into the assertion that P is a global at tractor (with respect to weak convergence) for the flow { Ut, t ___ 0 }, namely

(13) Ut#-"-~P, V # E M ,

where r Y �9 stands for weak convergence in M as defined in (3).

Just as in [Zakai, 1969, Thin. 3] we can prove the following result concerning these objects:

PROPOSITION 3.

Suppose each Pt(x,.) is equivalent to Lebesgue measure in X. Then

a) The invariant measure P is also equivalent to Lebesgue measure, and it is unique.


b) F o r every x E X and every f E LI(X,r P),

T

o x

This last assertion is most important in the following physical interpretat ion of the foregoing construction, given in the spiri t of Statistical Mechanics [Thompson, 1972].

Consider the motion of a particle along the trajectories of the Markov process {xt}, s tar t ing f rom an arbi t rary point x E X. This <~ particle >> can be thought of as an energy photon, moving at random within the tube. However, condition (13) means that an equilibrium condition is eventually reached, in which the particle's position within the bar can be described in probabilistic terms by P. Macro- scopically, this condition manifests itself as a uniform temperature profile along the tube, which is maintained in the sense of both (5) and (12). As to the ergodic property (14), it allows us to determine dynamical quantities depending on the particle's instant position by averaging them with respect to P, dispensing with the need to observe the particle for an indefinite length of time.

On the other hand, it is important to remark that this random behaviour encountered at a microscopic level manifests itself as a unidirectional, irreversible energy t ransfer when examined from a macroscopic perspective, as illustrated in section 2.

4. - A DIRECT CONSTRUCTION OF T H E PARTICLE'S TRAJECTORIES (d ~ - 1).

To simplify notation, wri te

a(x) : ~ 2a(x) , b (x ) : ~--- a ' ( x ) - - v (x) , x E X

in the B V P (2.8) and in all subsequent developments given in the preceding section. Assume the regulari ty condition

(la) a(x) _> ao > 0, x E X

holds, in addition to the continuity of both a and b. Thus, A is the closure of the differential operator given by

1 (lb) v ~-> -~- a (x ) v " -}- b(x) v"

40 D. B. HERNANDHZ

on C2(0, 1) N {v E C1[0, 1] : v'(0) ~ v ' ( 1 ) ~ 0}. Of par t icular interest is the special case a(x) ~-- 1, b(x) -~ 0, the corresponding process being known as brownian motion with reflecting barrier at 0 and at 1, (BMRB) due to the fo rm of its t ra jector ies : they a re continuous curves contained in the closed rectangle [0, ~o)X X, namely the graphs of all the continuous functions f rom [0, ~ ) into [0, 1]. Let { W~, x E X } be the solution to the submart ingale problem corresponding to the different ial operator

1 d~v 2 d ~ ' v ' ( 0 ) ~ v ' ( 1 ) ~ 0 ,

i.e. {W, , x E X} is the probabilistic description of brownian motion with reflecting barr iers a t both endpoints of X.

In fact, we have seen tha t for each choice of coefficients a, b in (lb), there is a family {P~, x E X} of probabili ty measures on (~2, A) which solves the submart ingale problem corresponding to A. Thus, solving this problem amounts to specifying a way to proba- bilize ~, i.e. the set of all continuous X-valued trajectories on [0, r and s tar t ing at each possible x E X. The corresponding process { xt , t _> 0 } is then known as a diffusion with reflecting barriers at both 0 and 1 (DRB). The operator A defined before is called the infinitesimal genera tor of the diffusion.

The following are two examples of measurable t ransformat ions f rom (~2, A) into itself:

EXAMPLE 1 (random t ime change).

Let ~ be a positive, continuous funct ion on X, and let (or, t __ 0) be the stochastic process on (~2, A) given by

t

o,(~) : ~ f ~ ( x , ( ~ ) ) ds , 0

for each t. Then, this new process has values in [0, ~), and it has str ict ly increasing trajectories. Let ( vt, t >__ 0 } be the process whose t ra jector ies are the inverse funct ions of the t rajectories of {vt, t ___ 0 }. Define the t ransformat ion T~co(t) :~(o(~t( (o)) , o~E~O for each t _ _ 0 . �9

DIFFUSION PROCESSES IN NONEQUILIRRIUM THERMODYNAMICS 41

EXAMFLE 2 (uniform value transformation).

Let ~ : X---> X be any Borel measurable transformation and let T~ be given by T~ co(t) : ~ ~(~o(t)), co E Y2, t _> 0. �9

Let T:~9-->~2 be any measurable transformation, and let {P~, x E X} solve the submartingale problem corresponding to A. T induces a new family of measures {Q~, x E X}, with

(2) Q~ :~--Pxo T -1 for each x EX.

It seems natural to pose the following questions:

Q1) does { Q ~ , x E X } solve a submartingale problem too? and, if so, which is the corresponding differential operator, say B?

More generally,

Q2) which are the measurable transformations T of ~9 which map solutions of a submartingale problem into solutions of (another) submartingale problem?

Partial answers to Q1 are provided by the following results, where it is assumed that {P~, x E X} solves the submartingale problem corresponding to the differential operator A as in (1), and {Q~, x EX} is defined as in (2), for a given measurable transformation T : ~9--> ~9.

PROPOSITION 4.

a) I f T ~---T ~ as in example 1, then {Qx, x E X} solves the submartingale problem for the differential operator B, wi th Bv : ~ Av/~.

b) I f T ~ T~ as in example 2, then {Qx, x E X} solves the submartingale problem for the differential operator B given by

v ~ ~'(~-l(x))~ a(~-l (x)) v" § A~(~-~(x)) v ' .

Assertion a) above constitutes Theorem 7.31 in [~ksendal, 1985]; assertion b) follows from Ito's differential rule combined with Theorem 7.24 of the same reference. Both theorems in [0ksen- dal, op. cit.] assume a representation of diffusion processes as weak solutions of appropriate submartingale problems, e.g. as in [Stroock-Varadhan, 1971].

42 D.B. HERNANDEE

As to Q2, it gets a partial answer in

PROPOSITION 5.

{P=, x E X} solves the submartingale problem corresponding to A and { Q~, x E X } solves the submartingale problema corresponding to B for T ~ T ~ o T~ i f and only i f

(3) A ( v o ~) = ~(Bv ~ ~) for every smooth v.

This is a consequence of Theorem 1 of [Csink-0ksendal, 1983], where the characteristic operator of the diffusion [0ksendal, 1985] is used insted of the infinitesimal generator. However, they coincide on C ~. The reader is referred to this source for a very general treatment of measurable transformations of the type T x o T~ .

On the other hand, Proposition 5 gives us the possibility of obtaining a given diffusion with reflecting barriers from a standard, one, say BMRB, by means of transformations of the type T~o T~ . Indeed, an easy computation shows that the right ~ ' s satisfy the differential equation

b(y) (4) y" = 2 y%

which can be readily solved for invertible solutions, giving

t b(y) dr t (5) f exp { - - 2 J - - ~ - ~ t~)dy:ClX--~.-C2,

with C1 > 0 and 6'2 arbitrary. For every ~ determined by the im- plicit relations (5), the right ~ is given by

o ' ( x ) : (6) - - ' x e X .

The foregoing development establishes the validity of the following result:

PROPOSITION 6.

Let q~ : X--> X be continuous and invertibIe, determined by (5) for a given choice of integration constants, and let 2 be as in (6). Then (and only then), T ~ o T~ transforms {Wx, x E X} into the D R B associated wi th the operator A as in (la, b).


5. - S M O L U C H O W S K I ' S D I F F U S I O N S .

Let us go back to model (28) for heat conduction in a moving fluid, in the part icular case of constant thermal diffusivity. Then, operator A in (3.8) becomes

d~u v(x) ~xx (u ~ C ~) (1) A u ~ . ~ dx-- ~

In the language of Stochastic Calculus [Schuss, 1980], the Markov process { xt, t >_ 0 } associated with A satisfies the stochastic differential equation

(2) dx ~ - - v (x) dt + ~-~ d W ,

where { Wt, t _> 0} is Wiener's process. I f A is restricted to the

domain C~(~:) N {u E CI(X) : u'l~x --- 0}, as required by the construc-

tion given in section 3, then equation (2) is satisfied for as long as

the process stays within .Y. In other words, letting ~ be the f i rs t

exit t ime of {x,, t _> 0} f rom ~ , then (2) holds in (r > t), for each

t __ 0. Ignoring these subtleties amounts to invoking Kac's r principle of not feeling the boundary >> [Kac, 1966] ; let us do just that and work directly with (2).

Recall the duality relation (3.10), as well as the physical interpretation of the stochastic model resulting thereof, allegedly given r in the spirit of Statistical Mechanics ~. However, the quoted sta- tement is not quite applicable, because in (2) there is neither mass nor acceleration. A mechanical stochastic model must somehow result f rom the basic equations of Mechanics i.e. f rom Newton's Second Law or Hamilton's equations for the particle system under consideration.

In 1965, Ford, Kac and Mazur (FKM) gave a deservedly famous construction whereby Hamilton's equations for a system of N ~-1 particles, labelled 0, 1, ..., N, give rise to

(3a) dx --- udt

(3b) mdu - - [ - - ~u ~- K(x)] dt -}- odW,

as a simplification valid when N--> ~. In (1) x and u denote the position and velocity of the zeroth particle, K(x) is a external force

44 D. B. HERNAND 1~.

and - - # u is a viscous force (~ > 0). In the F K M analysis the masses of the remaining particles go to zero as N--> ~.

Lately, Picci gave an analysis yielding the F K M construction as a byproduct, see [Picci, 1986]. In both the F K M and Picci's treat- ments, the external field is a harmonic oscillator, i.e. K ( x ) - - - coax. Model (3) was studied by Ornstein and Uhlenbeck in the 1930's (see [Uhlenbeck-Ornstein, 1930]) and the process { (xt,ut), t >_ 0} is therefore known as the Ornstein-Uhlenbeck model of brownian motion. On the other hand, a close analogue of (2) had been proposed by Smoluchowski as model for brownian motion too, namely

(aa) d x ~ T(f ~..,x, dt ~ ~/I/ 2kT d W ,

see [Smoluchowski, 1916]. Note that (2) corresponds to taking

(4b) K (x) ~ - - #v(x), a~u~- kT

in Smoluchowski's model, therefore (2) can be reconciled with Me- chanics if (4) is somehow shown to be obtainable from (3).

This will be done in what follows using both space-time scaling and asymptotic analysis, by combining both [Schuss, 1980, 6.1] and [Nelson, 1967]. See Proposition 4 below.

For, let us begin by remarking that each term in (3b) has action (i.e. force X time) units. On the other hand, it is well known that { Wt, t _> 0 } has quadratic variation over [a, b] equal to b - - a (see [Hida, 1980, Thm. 2.3], a property usually expressed in shorthand notation as <~ (dW) ~ ~ dt ~>. If t has time units, then each Wt must

have dimensions of I / ~ Therefore, ~ in (3b) must have units of

force X l/time. These considerations are important in order to check dimensional consistency throughout the following analysis.

It is shown in [Uhlenbeck-Ornstein, 1930] that both Ext and E ( x t - - E x ~ ) ~ evaluated from Smoluchowski's model (4) with K(x)-------coax are virtually indistinguishible from the same quantities evaluated from the Ornstein-Uhlenbeck model (3), provided friction is high and times are long. Let us present Nelson's quantitative result in what follows, whereby the so called SmoIuchowski 's approximation is firmly established, for general force field K.

Before we can do that, let us follow [Schuss, 1980, 6.1] and submerge (3) in an infinite family of models, one for each e > 0,


where e is a dimensionless parameter. These models are

dx = u dt

m d u ~ [ - - # s - l u + K(x)] dt + o~ d W ,

where o~ > 0 is to be specified. For, observe that if the Gibbsian principles of Equilibrium Statistical Mechanics are to be adhered to, equilibrium at absolute temperature T must obey the Maxwell- Boltzmann distribution [Thompson, 1972), i.e. that probability distribution with density

(5a) e(x, u ) =

I t (x ,u)

k T

H (8, y)

where the hamiltonian function H is given by

x

(5b) H(x, u) = mu2 f K(~) d~ 2

0

for this problem.

We know [Hernfindez-Pavon, 1989] that this is possible if and only if Einsten's relation holds, i.e.

2 eo ~ = 2#kT, E > O,

hence the family of models to be considered actually is

(6a) dx ~ udt

(6b) mdu ~-- [ - - s -1 #u + g(x)] dt + ] / ~ d W E

Let us change time scale by defining s : = st (hence both s

and t have physical units of time), and define the new processes

{x~,,s >_ 0 } , { u ; , s _ 0} and {W~, , s >_ 0} by means of x; : = x , / , ,

" r Ua : = e ' u ~ / ~ , W ~ : ~ / ~ .

It is a standard fact in the theory of the Wiener process that

{ W : , s > 0} is a Wiener process too (see [Hida, 1980, Pro. 2.1]),

46 D.B. HERNANDI~:

hence can be replaced by the original { W,, s >_ 0 }. With the above rescaling, the family of models (6) t r ans fo rms into

(7a) dx ~ ---~ u ~ ds

(7b) ~mdu �9 --- [ - - ~u" ~- K(x~)] ds -}- ~2~kT d W ,

hence a formal limit when e--> 0 leads to

(8) dx- - - K ( x ) ds ~ - | / 2 kT d W , v

i.e. to Smoluchowski's model (4).

To jus t i fy this formal construction it must be shown tha t

{ x; , s _> 0 } converges in a suitable sense to { x , , s >_ 0 } when e --> 0. This is established in [Nelson, 1967], whose Theorem 10.1 is re- produced below �9

PROPOSITION 7.

Let K be globally Lipschitz and let {Wt, t _ 0} be a Wiener pro-

cess. Le t {(x~,u ; ) , s _> 0} be the solution of (7), and let { x , , s >__ 0}

be the solution of (8), corresponding to x; ~ Xo, u~ ~ uo, wi th xo given. For any Uo,

lim

uni formly on compact subintervals of [0, ~), w i fh probabili ty 1.

According to Nelson's result, model (2) is but Smoluchowski's approximation to an Ornstein-Uhlenbeck particle in a force field K provided (4b) applies, and it is valid only for long times.

In general, the coefficient of the white noise te rm does not have to be constant, for instance in the case of heat conduction in non- homogeneous media. In such situations, the duali ty relation (3.10) will result in diffusions associated wi th stochastic di f ferent ia l equations of the general fo rm

dx ~--- b(z) dt ~- o(z) d W .


To give these diffusions (SmoIuchowski's, to distinguish them from the dynamical Ornstein-Uhlenbeck diffusions) a correct dynamical interpretation, it is necesary to extend the asymptotic analysis leading to Proposition 7. In turn, Einstein's formula must also be generalized to allow for variable o i.e. for media with nonhomo- geneous dispersion properties, possibly along the lines pursued in [Hern~ndez-Pavon, 1989]. See also [Okaba, 1985].

6. - E Q U I L I B R I U M AND TRANSIENT BEHAVIOUR.

Let us now go back to the mathematical developments derived from the duality relation (3.10). In the language of semigroup theory [Goldstein, 1985], A is the infinitesimal generator of the semigroup {Tt, t __ 0} and, from (3.10),

< T , f - - f t ' / ~ < f ' U , / ~ l * \ t /

It follows upon taking limits when t --> 0 that the adjoint A* of A is the infinitesimal generator of the adjoint semigroup { U,, t _< 0 }. See also [Goldstein, op. cir., Thin. 4.9]. Therefore

(1) u,,-, + f A*(U. t >_ 0 0

Letting ~ 6 M describe equilibrium (hence U, ~ ~, t >__ 0), clearly

t

( A*(Us-~)ds-----O,t >_ 0, 0

hence ~ must satisfy

(2) A* ~-= 0.

The converse is trivially true, hence ~E M is an invariant measure for the diffusion associated with A if and only if (2) is satisfied.

say If the stochastic process {x~, t _>_ 0} has a transition density,

P~(x, B) - - f p~(x, y) dy, B E ~23(X), B

48 D. B. HERNANDEZ

then

u, .(B)=f' .(a~) :_! p,(~, ~) a~/ X B

B X

and each Ut~ is absolutely continuous with respect to Lebesgue measure in X, with

dUt (3) - ~ - - ~ #(dx) pt (x, y) .

X

Recall tha t the adjoint operator A* of A is defined by means of

(4) ( A f , I~) ~ - ( f , A*#) , f E C(X) , 1~ E M .

In general, let us consider the different ia l operator

(5a) A/---af" + bf' + of,

with boundary condition of Neumann type, which we will rewri te as

(5b) - - a(0) f ' (0) = 0, a(1) f ' (1) ~ 0

Let # E M have a density q9 such tha t (a~)", (b~o)' and c~o are con- t inuous (as in [Feller, 1952]). Therefore,

1

( A f , la) ~ - / (af" + bf ' + el) cpdx 0

1 1

=a(:, ~) I + f :t(a~)"-(b~)' + ~ 1 e~, o o

where

J( f , ~o) : ~-- a(~of" - - f~o) -t- (b - - a') f~o;

the second equality is just if ied by a double integrat ion by parts, in a classical fashion; see [Stakgold, 1979, 2.5].

Let

(6) A*~o : ~ (a4o)" ~ (bqg)' + c~o


denote the formal adjoint of A and let B denote the r boundary operator >> given by

(7a) Bf(O) : ~ - - a(0) f '(0), Bf(1) : ~ a(1) f ' (1 ) .

then

1

(Af , la) ~ / f (x) A*~o(x) dx Jr [Bf(1) o

~p(1) - - Bf(O) ~o(0)]

- - f(1) [ - - a(0) ~o'(0) + b(0) ~(0)] + f(0) [a(1) ~'(1) -}- b(1) ~(1)]

and, in view of (4b), i t follows tha t

1

(Af , I~) = f f (x) A*~o(x) dx - - f(1) B*~o(0) q- f(0) B*~o(0), o

where

(7b) B*~(0) : ~ - - a ( 0 ) c#'(0) -1- b(0) cp(0), B*~o(1) : =

~ - a(1)q9'(1) + b(1)q~(1).

Upon restr ic t ing A* to smooth functions ~ sat isfying B*p ~ 0 whenever Bf ~-O, there results the equality

1

(Af , f~) ~ f f (x) A*cp(x)dx.

The following result then follows f rom (4):

P R O P O S I T I O N 8 .

Let A be the differential operator defined in (5) with its formal ad]oint given as in (6). I f # E 1~I has a density q~ such that A*~ /s continuous and B*p ~ 0, then A * : has a density, i.e.

A*~(B) ~ f A*o(x) dx [B E B(X)] . B

Taking into account the foregoing result as well as (3), the

S e m l n a r i o M a t e ~ n a t i e o e F i ~ i c o - 4

50 D . B . HERNANDE~

integral equation (1) t ransforms into

t

i <.,, ., .>_.o<., .> - / A...<., ., . > .B 0

for each # E M, each B E B(X). Therefore

OP, ~ A ' p , (8a) ~t [t > O, x E X]

besides

(8b) B * p t ( x , . ) ~ O , t >_ 0

This partial differential equation is known as Fokker-Plank 's and, together with the boundary condition (8b), it must be satisfied by the transit ion density as a function of (t, y), for each value of x.

In particular, the equilibrium condition (2) specializes into the s tat ionary Fokker-Planck boundary value problem

(9a) A * p ~ 0 [x E X]

(9b) B*p(z , . ) ~ 0

to be satisfied for each fixed x E X.

These two Fokker-Planck boundary value problems constitute the main tools for the study of equilibrium and stability in terms of the stochastic process associated with every irreversible process, as described in section 3 above.

Note that the ordered pairs of operators (A, B) and (A*, B*) are equivalent, in the sense tha t the corresponding boundary value problems yield the same information on a given irreversible process. The f i rs t pair appears in deterministic macroscopic formulations such as those encountered in section 2; the second pair appears in the stochastic, microscopic formulation obtained in section 3. In a given situation it may be more convenient to use (A, B) in order to study equilibrium and stability of a given stochastic system (like the Ornstein-Uhlenbeck process, for instance). Or else it may be advisable to use (A*, B*) in order to obtain equilibrium and stability information on a given irreversible thermodynamic system.

DX~FUSION ~OCESS~S IN NON~.qurL~RmM T ~ M O D r ~ A m C S 51

Let us illustrate this duality as follows. There are two main goals in the study of stability and equilibrium in a stochastic system:

a) finding the invariant measure of the process,

b) determining the rate of approach to equilibrium.

According to what has been said, one may either solve

(10) A * ~ 0 in .~, B*~]ax----0

for the invariant measure or else determine all the equilibrium

states u satisfying

(11) A u - - 0 in X, B u l a x - - 0

and then determine Q such that

g ~ f f(x) ~(x) dx, for suitable f . X

Therefore a) can be achieved in either fashion, but presumably solving (10) is the easier way.

To achieve b), one may decide to solve the initial value problem

(12a) Ou at ~ A u in ( 0 , ~ ) X

(12b) B u ( t , . ) [ ~ x ~ O , t >_ 0

(12c) u(O,.)~---f .

The solution of (12) obtained by separation of variables is

o o

~tkt (13) u(t, x) ~ (f) q- Z c~ e ~ ( x ) ,

where 0 ~ ~o > 2~ > ... is the spectrum and {~} is the sequence of eigenfunctions of the problem

A y - - ,ty in X, By] 0x ~ 0.

The eigenfunctions are orthogonal with respect to the inner product

52 D. B. HERNANDEZ

generated by a certain weight function e (the integrating factor), and

(14) (f) ~- f f(x)~(x) dx. X

Therefore ~ solves a). Moreover, it is clear from (13) that

- - e ~ l t u(t , X ) - - U = C l ~1(~) [1 q- o(1)] ,

hence the rate of approach to equilibrium is given by the first negative eigenvalue ~1 of operator A.

TO conclude, it is important to remark that the weight e in (14) is the density of the required invariant measure. In fact, it is always true that an invariant measure exists when the state space of the process is compact [Kunita, 1971, 3.2], but the previous remark offers a constructive method in order to find it; summing up: obtaining invariant measures for a given diffusion process amounts to giving integrating factors for its infinitesimal generator. In particular the invariant measure of a diffusion processes with reflecting boundaries and a selfadjoint infinitesimal generator is the uniform distribution in its state space.

SUNTO. - - La teoria dei semigruppi consente il collegamento fra due for- mulazioni alternative in termodinamica irreversibile: una, macroscopiea e de- terministica; l 'altra, microscopiea e stocastica. Tramite questa associazione si ricava che le densit~ invarianti per una diffusione sono i fattori integranti del suo generatore infinitesimale. Inoltre, sussiste una propriet~ ergodica, e le propriet~ macroscopiche si ricavano dalla formulazione microscopica tramite le rap- presentazioni probabilistiche delle soluzioni di appositi problemi con valori ini- ziali e al contorno.

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Diffusion processes in nonequilibrium thermodynamics

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