BUDAPEST PHYSICS INSTITUTE FOR RESEARCH CENTRAL

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L, D I Ó S I B. L U K Á C S

C O V A R I A N T E V O L U T I O N E Q U A T I O N F O R M A L I S M F O R T H E T H E R M O D Y N A M I C F L U C T U A T I O N S

H ungarian cAcademy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

2017

н

••

COVARIANT EVOLUTION EQUATION FORMALISM FOR THE THERMODYNAMIC FLUCTUATIONS

L. DIŐSI, B. LUKACS

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

HU ISSN 0368 5330 ISBN 963 372 260 8

ABSTRACT

A system of equations is given for the distribution of the fluctuations of arbitrary thermodynamic state variables, by exploiting the Riemannian structure of the thermodynamic state space. These equations have been made compatible with the Second Principle of Thermodynamics, and for small fluc­

tuations they reproduce the usual Gaussian law. We show a real stochastic process resulting in these equations.

АННОТАЦИЯ

С помощью Римановой структуры пространства термодинамических состояний пыподитси система уравнений для распределения флуктуаций любой термодинами­

ческой величины. Полученные уравнения автоматически являются совместимыми со вторым началом термодинамики, и для маленьких флуктуаций они воспроизводят обычное гауссовское распределение. Показывается действительный стохастичес­

кий процесс, приводящий к атим уравнениям.

KIVONAT

A termodinamikai állapottér Riemann-strukturáját felhasználva megadunk egy egyenletrendszert tetszőleges termodinamikai állapothatározó fluktuá­

cióinak eloszlására. Az egyenletek konstrukciójuknál fogva tiszteletben tartják a Második Főtételt, és kis fluktuációkra visszaadják a szokásos Gauss-eloszlást. Mutatunk olyan valódi stochasztikus folyamatot, mely ezen egyenletekre vezet.

1. I N T R O D U C T I O N

Statistical mechanics'*“ yields a f o u n dation of thermodynamics, w h i c h latter is obta i n e d in the s o - c a l l e d the r m o d y n a m i c limes,

i.e. when the size of a given h o m o g e n e o u s equi l i b r i u m s y s t e m goes to infinity. Then, of course, the fluct u a t i o n s of the t h e r m o ­ dyn a m i c characteristics vanish.

When the s y s t e m is m a c r o s c o p i c but finite, fluctuations appear in it w i t h some p r o b a b i l i t y distribution, and t h ere exist thermodynamic e x p r e s s i o n s for t h eir first a n d second m o m e n t a ' . 2 3 Thus, although the thermodynamic limit c a n n o t yield all the in­

formations a b out finite systems, one may e x p e c t the r m o d y n a m i c formulation for the distr i b u t i o n of the fluctuations. In fact, various d istributions can be c o n s t r u c t e d w i t h differ in h i gher momenta; the first and simplest o n e was p r o p o s e d by E i n s t e i n . 2 Here we want to formulate an e q u a t i o n for these distributions, w h i c h are s l i g h t l y mo r e complicated, but pos s e s s c e r t a i n a d van­

tageous properties.

2. E V O L U T I O N E Q U A T I O N F O R T H E E X T E N S I V E F L U C T U A T I O N S

Consider a h o m ogeneous e q u i l i b r i u m s y s t e m of infi n i t e volume. The t h e r m o d y n a m i c state of this s y s t e m is c o m p l e t e l y determined by the set of n i n d ependent e x t e n s i v e d e n s i t i e s

{x ,i 1 , 2 , . . . , n } .

Take a s u b s y s t e m of finite v o l u m e V. F o r it, the r e m a inder of the infinite system is a reservoir. D e n o t e the s t ate of the reservoir by x ^ and that of the finite s u b s y s t e m by x 1 . O b v i ­ ously, x wi l l fluctuate around ^{x q} w i t h cer t a i n p r o b a b i l i t y Pv (x|xQ )dnx.

2

W h e n the s u b s y s t e m is infinite t o o there are no f l u c t u a ­ tions, i.e.

p (xIx ) = 6 ( n ) (x-x ) . (2.1)

I о О

We are l ooking for the function p v (x |xq ). F i r s t we d e r i v e a set of c o n straints for it from the S e c o n d Principle of T h e r m o - dynamics . Obviously, the e x p e c t a t i o n values of x s mus t be in­4 d e p e ndent of V, thus equal to the r e s e r v o i r value:

otherwise, after a fictitious s e p a r a t i o n of a g r e a t n u m b e r of subsystems and r e j o ining the m in a g r e a t e r subsystem, this latter one w o u l d not be in m a c r o s c o p i c e q u i l i b r i u m wit h the reservoir.

The S e cond P r i n c i p l e forbids this.

Q

Now, there exists an a p p r o x i m a t i v e p (x|x ) for p (x|x ) if

2 v о v о

the fluctuations are small :

PV (x|xo }

n / 2 ----

) v/|g (x ) Iexp{- v n 2 I

i ,k=l

. . , i i, , к к. , g ,, x ) (x -x ) (x —x ) }

^ik о о о

(2.3) Here

g i k <x)

(x)

„ i. к ' Эх Эх

г,к 1,2, . . . ,п , (2.4)

and I g I denotes the dete r m i n a n t of g^^,-

T h ere is no reason to b e l e i v e the form (2.3) if x is far Q

from x since g,, is taken at x thus the d i s t r i b u t i o n pt7 is not

о ^ik о

affected by the g l o b a l p r o p erties of the state space.

N e v e r t h e l e s s the leading terms s e e m correct, thus let us start w i t h the G a u s s i a n a p p r o x i m a t i o n of p ^ (x|xQ ). I n t roducing a "time" variable x,x=l/V ins t e a d of V, the G a u s s i a n distribu-

G G

tion p J(x,x|xo ) = p v (x |xq ) fulfils a diffusion type e v o l u t i o n e q u a t i o n :

Э Эх

p (x,xG /

x o } =

i k . >

g x ) о

G. I

P (t,x | Xo )

» i, 1c Эх Эх

(2.5)

3

w h e r e g ik is the in v e r s e m a t r i x of g ^ , w i t h the initial c o n d i ­ tion

p G (0,x|x ) = (x-x ) . (2.6)

c о о

(Henceforth we adopt the E i n s t e i n convention: there is a s u m m a ­ tion if an index o c c u r s twice, above and below.)

As we m e n t ioned above the evolution e q u a t i o n should co n t a i n the local structure of the t h e r m o dynamic state space even for x's w h i c h are far from the initial ^xq . In o r d e r to e n s u r e this p r o p ­ erty we have to g e n e r a l i z e the Gaussian e q u a t i o n (2.5).

i к j к

We should use g (x) in s t e a d of g (xq ) t h e ec3* (2.5) and complete this ex p r e s s i o n w i t h terms c o n t a i n i n g the d e r ivatives of

i к

g . We have to get total div e r g e n c e on t h e rhs and to satisfy the constraints (2.2) as well. The only p o s s i b l e choice is then

2

£ p(T,x|x ) = ^— 7- g ik (x) р(т,х I x ) . (2.7)

О L О гчХп.К О

dX dX

This gene r a l i z e d e v o l u t i o n equation and the initial c onti- tion

p ( O fx|x ) = 6 ^ n ^(x-x ) (2.8)

1 о о

are suggested to d e s c r i b e the p r o b a b i l i t y d i s t r i b u t i o n of the t hermodynamic fluctuations in a given s u b s y s t e m of finite v o l u m e V = 1 /t.

C o n s i d e r finally the case when we are not interested in the d i s t r i b u t i o n of the n'th e x t e n s i v e x n , looking for the d i s t r i b u ­ tion p (t ,x |xq ) of x: ( x \ x ^ , . . . ,xn ■*■) :

p ( r , x | x o ) = p (T,xI^xq )dxn

(2.9) If one integrate the e q s . (2.7,8) b y d x 11 it can be shown that p ( x , x | x Q ) obeys the same evolution e q u a t i o n as p (t,x |xq ) does, w i t h the substitutions: n -»• n-1, x -»■ x.

4

3. E V O L U T I O N E Q U A T I O N F O R T H E F L U C T U A T I O N O F G E N E R A L V A R I A B L E S

In the previous Section an equation w a s found f o r the

fluctuations of extensive densities, w h ich 1) is of quite n atural form, 2) fulfils the Second P r i n c i p l e of T h e r m o d y n a m i c s , and

3) yields the Gaussian a p p r o x i m a t i o n for large V.

Now, in many cases one is interested in the fluctuation of other quantities (as e.g. the intensives) . Of course, these q u a n ­ tities are functions of the e x t e n s i v e densities, so the f luctua­

tion p r obabilities can be d e t e r m i n e d through pv (x|xQ ). Denote the extensive densities by x 1 and a general complete set of other parameters by a:1 , then

x^ = cp* (ж) , i = l , 2 , . . . ,n . (3.1)

This is a coordinate t r a n s f o r m a t i o n on the state space.

Obviously the distri b u t i o n of the new v a r i a b l e s w i l l be

P v U | V = PV ( x lXo }

Эф1 (a?) а к

(3.2)

Nevertheless, there is an o ther way too, n a m e l y to find an evolution equation directly for p (ac | a: ) , and one has to see if the two P y fs are the same.

We a d opt the idea that the the r m o d y n a m i c state space is a Riemannian m etric space. The m e t r i c tensor in e x t e n s i v e coordi­

nates is def i n e d by eq. (2.4) D ' . In order to get a unique d i s ­ tance in a coordinate t r a n s f o r m a t i o n (3.1) g t, m u s t change

—j IK according to the usual t r a n s f o r m a t i o n law

9 l k U ) = g (x) 3rs

ЭфГ (д?) Эф5 (Я)

(3.3)

Now, we ne e d a covariant e v o l u t i o n equation for p (ж|а: ),

because covariance a u t o m a t i c a l l y guarantees that the quantities change p r o p e r l y wi t h the c o o r d i n a t e transformation, so then

5

Ру(д:|хо ) will be unique. Naturally, this c o v a riant equation has to possess the form (2.7) in extensive coordinates. Since the R i e mannian geometry 7 has a p roper covar i a n t formalism, henceforth we adopt its method.

Note first that р ^ ( ж | ж о ) is not a s c a l a r , s i n c e the covariant volume element is / 1 g Í d n.r. Thus the s calar q u a n t i t y is p,

р(т,ж|ж ) = --- ---- р „(ж|ж ) . (3.4)

о /г ",— гг V ' о

/\д(х) I

O b v i o u s l y the initial c o n d ition (2.8) is now

0(0,*1* ) = 1 S (n) ( x-x) . (3.5)

° / | g ( * 0 )|

The form of eq. (2.7) shows that we need a diffusion type equation for p. We claim that the p r o p e r form is

тг~~ р(т,ж|ж ) = Др(т,ж|ж ) + V (hr (ж) p (т ,x I ж )) (3.6)

d г о о г о

w h ere V. stands for the c o v a riant derivative, Д is the covariant

1 7 i

L a p l a c i a n , and h is a v e c t o r field g u a r a n t e e i n g constraints

(2^.2^):

Ф (ж) p (t ,a: I ж ) /| g (ж) | d ^ = cp1 I (ж ) (3.7) Here the bar in cp1 denotes that i is not a v e c t o r i a l index but a n a m e .

By differentiating this equation w i t h r espect to x, using eq. (3.6), and per f o r m i n g par t i a l integrations, one gets

' (Д ф 1

J h r ) p/| g I d ^ 0 . (3.8)

Since this equation mus t h o l d for any p, the r esult is

(3.9)

6

A c c o rding to eq. (3.1), cp'1 ' occurs in a c o o r dinate transforma- x I к

tion, so Эср 1 /Эх m u s t possess a regular m a t r i x and the vector field h 1 is then uniquely d e t e r m i n e d by eq. (3.9).

E q s . (3.6), (3.9) are covariant. In e x t e n s i v e coordinates

eq. (3.9) takes the form *»

hi

эг (дг з /Тд|а5Ф 1 ) (3.10)

w h i l e eq. (3.6) can be w r i t t e n as

/I

3 /[i

(3.11)

p = /|g|p •

By combining eqs. (3.10) and (3.11) they reduce to eq. (2.7).

Thus eqs. (3.4-6), (3.9) are the covariant forms of the evolu­

tion equation (2.6-7).

4. S T O C H A S T I C F O U N D A T I O N O F T H E E V O L U T I O N E Q U A T I O N

In the previous two sections we p r o p o s e d a new evolution equation which w o u l d govern the d i stribution of the t h e r m o ­ d ynamic fluctuations arising in finite e q u i l i b r i u m systems. An e legant form of covariant d i f f usion equation on the Riemann m e t r i c i z e d state space was found.

Here we are going to show that this d i f f u s i o n comes from a true stochastic process w h i c h is accomp l i s h e d on the state space.

Originally, w h e n R u p p e i n e r 5 supplied t h e r m o d y n a m i c state space wit h Riemannian m e t r i c (2.4) he also outli n e d a stochastic process. But he did not vary the volume of his s y stem thus the physical realization of his construction was not quite clear.

Consider a homogeneous closed e q u i l i b r i u m s y stem Q(V,x) of

X 2 n

volume V and extensive densities x = ( x x ,.. . ,xa ) . Now, at random, let us choose and separate a s u b s y s t e m fl(V',x') in it:

£3 (V' ,x') CQ(V,x) , V ' < V , (4.1)

7

and denote the prob a b i l i t y of finding x' at a given v a l u e by P ( V ' , x 7 |v,x)dn x ' .

O b v i o u s l y ,

P (V , x 7 IV,x) = 6 (n) (x'-x) . (4.2)

Repeating the above p r o c e d u r e and choose a subsystem Q ( V 7 7 , x 77) in Q ( V 7, x 7), we get the cond i t i o n a l d i s t r i b u t i o n P (V' 7 , x 7 7 IV' , x ') for x " . It can o b v i o u s l y be supposed that the final distri b u t i o n of x 7 7 , i.e.

P ( V 7 7 , x 7 7 I V 7 , x 7) P ( V 7 , x 7 |v,x)dn x 7 (4.3) is not affected by the interm e d i a t e s e p a ration of Q ( V 7 , x 7).

T h e r e f o r e the prob a b i l i t y (4.3) m u s t be equal to the p r o bability d i s tribution of x 77 in a s u b s y s t e m fi(V7 7 ,x77) w h ich was directly c hoosen from fl(V,x):

P(V 7 7 ,x 7 7 I V 7, x 7)P ( V 7 , x 7 |v,x)dnx 7 = P ( V 7 7 , x 7 7 |V,x),

V 77^ V 7<V . (4.4)

Equations (4.2), (4.4) show that the pro c e s s of continuously d i m inishing the volume of a h o m o g e n e o u s e q u i l i b r i u m s y s t e m can be

g

c o n sidered as a continuous M a r k o v i a n stochastic process . The role of stochastic variable is p l a y e d by the state c o o r dinate x of the actual system. The v olume reduc t i o n m u s t be adiabatic, i.e. slower than the r e l a xation of the t h e r m o dynamic f l u c t u a ­ tions .

Now let us suppose th a t this stochastic process is of finite vari a n c e and thus it is g o v e r n e d by the F o k k e r - P l a n c k - K o l m o g o r o v

g differential equation .

In order to find the coeff i c i e n t s of the d i f f e r e n t i a l equation we have to calculate the following two limits:

- , i i,

< x7 - X -

V - V 7

x 7l- x 7

P ( V 7 , x 7 IV,x)dn x 7 lim

V'->V

lim V'+V

(4.5)

8

. ,i. i w ,k 1

lim - x ) ( X ' X -

v'-*-v v-v' lim

V ' -»-V

. ,1 x. , ,k к

(x -x ) (x -x , |TT , ,n , --- v v 1--- -P(V',x' V , x ) d x'

(4.6) The expression (4.5) is o b v i ously zero since the S econd Principle requires that

<xГi

(4.7)

As for the ex p r e s s i o n (4.6), w e should k n o w the corr e l a t i o n of the extensive densi t i e s x' ’’ of the s u b s y s t e m Q (V' ,x') cQ (V,x) . The problem is that in (4.6) V'+V and thus Q(V',x') c annot be a small subsystem of Q(V,x). But we can easily a void this trap vising the c o mplementary system Q(V,x) = Q ( V , x ) \ Q ( v ' , x ') w h i c h is already a small subsystem. It is k n o w n that the c o r relation of the extensive d e n s ities x 1 in such a small s u b s y s t e m is g i ven as f o l l o w s ^ '^ :

< ( x * - x S ( x k - x k )> = - 3 g*"k (x) (4.8) V

Jlк

w h ere g (x) is the inverse of the m a t r i x g ^ ( x ) / see d e f i nition (2.4) .

Now using the bal a n c e equation

Vx + (V-V)x' = Vx (4.9)

we can eliminate x' from the expression (4.6) and we get instead:

lim — (xi- x i ) (xk -xk )\ = - -4r g i k (x) . (4.10)

v+0 v s v “v; 7 v

This e x p r e s s i o n is the coefficient function in the Fokker- Planck-Ko l m o g o r o v equation:

9V^T P(V',x'IV,x) = H - 4 g l k ( x ' ) P ( V ' , x ' | v , x ) ) . (4.11) 9 х ,3-Эх'К

V VZ

9

In the v a r i a b l e x(x=l/V) the same equation is of the f o l l o w ­ ing form:

2

4 t P(t',x'|t,x) = --- г---T- (gl k (x')P(T',x' |x,x) ) (4.12)

d T Э х # 1 Э х , К

and the initial c o n d ition (4.2) wi l l be as

P(x,x'|x,x) = 6 ^ n ^(x'-x) . (4.13)

It is just the p roper m o m e n t to note that the d i s t r i b u t i o n Pv (x |xq ) defined in Section 2 c o r r e s p o n d s obviously to the

transition p r o b a b i l i t y P(V,x|°°,xo ) and thus p ( x , x | x Q ) of Section 2 is equal to P ( x , x | 0 , x o ) in v a r i a b l e x. Therefore equations

(4.12), (4.13) y i e l d the e v o l ution equat i o n s (2.7), (2.8) which w e r e introduced in a formal w a y in S e c t i o n 2, and also the

covariant form (3.6), (3.9) of the e v o l u t i o n equation gets its stochastic foundations.

5. C O N C L U S I O N

We have p r o p o s e d a covariant s ystem of equations for the fluctuations of t h ermodynamic characteristics. For small fluc­

tuations they r e p r o d u c e the usual G a u s s i a n distribution, and from practical v i e w p o i n t the d i f f e r e n c e is small c o m p a r e d to any other possible d i s t r i b u t i o n too. Nevertheless, bec a u s e of the covariance our e q u a t i o n s can be d i r e c t l y used in arbitrary variables. In addition, the S e c o n d P r i n c i p l e of T h e r m o dynamics automatically holds too. The e q u a tions p r e s e n t e d here are the simplest possible ones possessing these t h eoretically important p r o p e r t i e s .

IO

R E F E R E N C E S

1. L.D. Landau and E.M. Lifshits, Stat i s t i c a l Physics (Pergamon, London-Paris, 1958).

2. A. Einstein, Ann. d. P h y s . 3^3, 1275 (1910).

3. R.F. Greene and H.B. Gallen, Phys. Rev. 8_3, 1231 (1951).

4. H.B. Callen, T hermodynamics (Wiley, N e w York, I960).

5. G. Ruppeiner, Phys. Rev. A 2 0 , 1608 (1979).

6. L. Diósi et al., Phys. Rev. A29 (1984) to be published.

7. L.P. Eisenhart, Riemannian Geometry (Princeton University, Princeton, N.J., 1926).

8. B.V. Gnedenko, The Theory of Prob a b i l i t y (Mir Publishers, M o s c o w , 1969).

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Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Szegő Károly

Szakmai lektor: Perjés Zoltán Nyelvi lektor: Dolinszky Tamás Gépelte: Végvári Istvánná

Példányszám: 190 Törzsszám: 84-391 Készült a KFKI sokszorosító üzemében Felelős vezető: Töreki Béláné

Budapest, 1984. junius hó