MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 18 doc

During the years 1920- 1930 Wiener approaches Brownian motion in terms of path integrals.. 20.11 .la_ From our discussion of Green’s functions in Chapter 19 we recall that W X , t, XO,

20 GREEN'S FUNCTIONS

In 1827 Brown investigates the random motions of pollen suspended in wa- ter under a microscope The irregular movements of the pollen particles are due to their random collisions with the water molecules Later it becomes clear that many small objects interacting randomly with their environment behave the same way Today this motion is known as Brownian motion and forms the prototype of many different phenomena in diffusion, colloid chem- istry, polymer physics, quantum mechanics, and finance During the years 1920- 1930 Wiener approaches Brownian motion in terms of path integrals This opens up a whole new avenue in the study of many classical systems

In 1948 Feynman gives a new formulation of quantum mechanics in terms of path integrals In addition to the existing Schrodinger and Heisenberg formu- lations, this new approach not only makes the connectlion between quantum and classical physics clearer, but also leads to many interesting applications in field theory In this Chapter we introduce the basic features of this technique, which has many interesting existing applications and tremendous potential for future uses

20.1 BROWNIAN MOTION AND THE DIFFUSION PROBLEM

Starting with the principle of conservation of matter,

equation as

we can write the diffusion

(20.1)

633

634 GREEN’S FUNCTIONS AND PATH INTEGRALS

where p ( T ‘ , t ) is the density of the diffusing material and D is the diffusion constant, which depends on the characteristics of the medium Because the diffusion process is also many particles undergoing Brownian motion a t the same time, division of p ( 7 , t ) by the total number of particles gives the probability, w(+,t), of finding a particle at 7 and t as

l i m w ( 7 , t ) + S(?) (20.4)

t-0

In one dimension we write Equation (20.3) as

(20.5) and by using the Fourier transform technique we can obtain its solution as

WIENER PATH INTEGRAL APPROACH TO BROWNIAN MOTION 635

satisfying the initial condition

lim W ( X , t, zo, to) -+ S(X - ZO) (20.10) t-to

and the normalization condition

&W(z, t, X o , to) = 1 (20.11)

.la_

From our discussion of Green’s functions in Chapter 19 we recall that

W ( X , t, XO, to) is also the propagator of the operator

(20.12) Thus, given the probability a t some initial point and time, w(z0, to), we can find the probability at subsequent times, w ( z , t), by using W ( z , t , so, to) as

in Brownian motion as in the Huygens-Fresnel equation

20.2 WIENER PATH INTEGRAL APPROACH TO BROWNIAN

636 GREEN’S FUNCTIONS AND PATH INTEGRALS

fig 20.1 Paths C[zo,o,to;z,t] for the pinned Wiener measure

Assuming that each step is taken independently, we combine propagators N times by using the ESKC relation to get the propagator that takes us from (20, to) to (z, t ) in a single step as

This equation is valid for N > 0 Assuming that it is also valid in the limit as

N -+ 00, that, is as At; -+ 0, we write

Here, T is a time parameter (Fig 20.1) introduced to parametrize the paths

as ~ ( 7 ) We can also write W(z, t, zo,to) in short as

W(z,t,zc),to) = Njexp{-& p ( T ) d T } i)z(7), (20.20)

WIENER PATH INTEGRAL APPROACH TO BROWNIAN MOTION 637

where N is a normalization constant and Dx(T) indicates that the integral should be taken over all paths starting from (z0,to) and end a t (z,t) This expression can also be written as

W ( z , t, zo,to) = 1 &&), (20.2 1)

C [ z o t o ; ~ , t l

where d , z ( ~ ) is called the Wiener measure Because dwz(r) is the measure for all paths starting from (zo, to) and ending a t (z, t ) , it is called the pinned

(conditional) Wiener measure (Fig 20.1)

Summary: For a particle starting its motion from (zo,to), the propagator

W ( z , t , zo, to) is given as

This satisfies the differential equation

with the initial condition limt4to W(z,t, zo, to) + b(z - zo)

In terms of the Wiener path integral the propagator W ( z , t, 20, to) is also expressed as

W ( z , t, 20, to) = .i’ d W 4 7 ) (20.24)

C [ ~ O , t O ; Z J l

The measure of this integral is

Because the integral is taken over all continuous paths from (20, to) to

(3, t ) , which are shown as C[zo, to; z, t], this measure is also called the pinned Wiener measure (Fig 20.1)

For a particle starting from (zo,to) the probability of finding it in the interval Ax a t time t is given by

(20.26)

In this integral, because the position of the particle at time t is not fixed,

d , z ( ~ ) is called the unpinned (or unconditional) Wiener measure At

638 GREEN’S FUNCTIONS AND PATH INTEGRALS

Fig 20.2 Paths C[zo,lo;t] for the unpinned Wiener measure

time t, because it is certain that the particle is somewhere in the interval

z E [-oo,oo], we write (Fig 20.2)

The average of a functional, F[z(t)], found over all paths C[zo, to; t] at time

t is given by the formula

In terms of the Wiener measure we can express the ESKC relation as

(20.28)

THE FEYNMAN-KAC FORMULA AND THE PERTURBATIVESOLUTION OF THE BLOCH EQUATION 639

20.3 T H E FEYNMAN-KAC FORMULA AND T H E PERTURBATIVE

SOLUTION OF T H E BLOCH EQUATION

We have seen that the propagator of the diffusion equation,

aw(z,t) a2w(z, t ) = 0,

can be expressed as a path integral [Fq (20.24)] However, when we have a

closed expression as in Equation (20.22), it is not clear what advantage this

new representation has In this section we study the diffusion equation in the

presence of interactions, where the advantages of the path integral approach

begin to appear In the presence of a potential V(z), the diffusion equation

where wo(z, t ) is the solution of the homogeneous part of Equation (20.31),

that is, Equation (20.5) We can construct WD(Z, t, z’, t’) by using the p r o p

agator, W(z, t , z’, t’), that satisfies the homogeneous equation (Chapter 19)

(20.33) is still not the solution, that is, it is just the integral equation version

of Equation (20.31) On the other hand, WB(Z, t, x’, t’), which satisfies

640 GREEN'S FUNCTIONS A N D PATH INTEGRALS

The first term on the right-hand side is the solution of the homogeneous equation [Eq (20.34)], which is W However, because t > to we could also write it as W,

A very useful formula called the Feynman-Kac formula (theorem) is given as

1

t

W B ( Z , t, Zo, 0) = J' ci,z(T) exp { - ciTv[x(T), 7-1 (20.38)

This is a solution of Equation (20.36), which is also known as the Bloch equation, with the initial condition

we integrate over the intermediate x variables and rearrange to obtain

W B ( Z , t , xo, to) = W(x, t, xo, to) (20.43)

j=1

DERIVATION OF THE FEYNMAN-KAC FORMULA 641

In the limit as E -+ 0 we make the replacement E X ~ -+ h”,tj We also suppress the factors of factorials, (l/n!), because they are multiplied by E ~ ,

which also goes to zero as E -+ 0 Besides, because times are ordered in Equation (20.43) as

we can replace W with WD in the above equation and write WB as

(20.44)

Now WB(z,t,~o,tO) no longer appears on the right-hand side of this equa- tion Thus it is the perturbative solution of Equation (20.37) by the itera- tion method Note that W~(x,t,xo,to) satisfies the initial condition given in Equation (20.39)

We now show that the Feynman-Kac formula,

is identical to the iterative solution to all orders of the following integral equation:

which is equivalent to the differential equation

with the initial condition given in Equation (20.39)

We first show that the Feynman-Kac formula satisfies the ESKC [Eq (20.14)] relation Note that we write V[Z(T)] instead of V[Z(T),T] when there

642 GREEN'S FUNCTIONS AND PATH INTEGRALS

(20.48)

In this equation x, denotes the position at t, and x denotes the position a t t Because C[ZO, 0; x,, t,; z, t] denotes all paths starting from (xo,O), passing through (x,,t,) and then ending up at (x,t), we can write the right hand-side

of the above equation as

Kac formula satisfies the initial condition

INTERPRETATION OF v ( X ) IN THE BLOCH EQUATION 643

The first term on the right-hand side is the solution of the homogeneous part

of Equation (20.36) Also, for t > 0, we can write WD(ZO,O,Z,~) instead of W(zo,O, z, t) Because the integral in the second term involves exponentially decaying terms, it converges Thus we interchange the order of the integrals

to write

(20.56) where we have used the ESKC relation We now substitute this result into Equation (20.55) and use Equation (20.45) to write

20.5 INTERPRETATION OF V(z) IN THE BLOCH EQUATION

We have seen that the solution of the Bloch equation

with the initial condition

WB(z,t,ZO,tO)lt=to = S(z - X o ) ,

is given by the Feynman-Kac formula

(20.60)

644 GREEN’S FUNCTIONS AND PATH INTEGRALS

In these equations, even though V ( x ) is not exactly a potential, it is closely related to the external forces acting on the system

In fluid mechanics the probability distribution of a particle undergoing Brownian motion and under the influence of an external force satisfies the differential equation

(20.62) where ?;I is the friction coefficient in the drag force, which is proportional to the velocity In Equation (20.62), if we try a solution of the form

-

we obtain a differential equation to be solved for W ( z , t; X O , to):

where we have defined V ( x ) as

1 1 d F ( x )

V ( x ) = - - P ( Z ) 4q2D + 2?;1 d x (20.65) Using the Feynman-Kac formula as the solution of Equation (20.64), we can write the solution of Equation (20.62) as

INTERPRETATION OF v(Z) IN THE BLOCH EQUATION 645

and used Equation (20.65)

As we see from here, V ( z ) is not quite the potential, nor is L[z(r)] the Lagrangian In the limit as D i 0 fluctuations in the Brownian motion disappear and the argument of the exponential function goes to infinity Thus only the path satisfying the condition

or

(20.72)

(20.73) contributes t o the path integral in Equation (20.70) Comparing this with

m, = -772 + F ( z ) , (20.74)

we see that it is the deterministic equation of motion of a particle with negli- gible mass, moving under the influence of an external force F ( x ) and a friction force -72 (Pathria, p 463)

When the diffusion constant differs from zero, the solution is given as the path integral

(20.75)

In this case all the continuous paths between (zo,to) and (z,t) will contribute

to the integral It is seen from equation Equation (20.75) that each path contributes t o the propagator W ( z , t, 20, to) with the weight factor

646 GREEN’S FUNCTIONS AND PATH INTEGRALS

Naturally, the majority of the contribution comes from places where the paths with comparable weights cluster These paths are the ones that make the functional in the exponential an extremum, that is,

6 lot dTL[X(T)] = 0 (20.76) These paths are the solutions of the Euler-Lagrange equation:

- - P [ d L d L 1 4

At this point we remind the reader that L[E(T)] is not quite the Lagrangian

of the particle undergoing Brownian motion It is interesting that V(z) and L[z(T)] gain their true meaning only when we consider applications of path integrals to quantum mechanics

20.6 METHODS OF CALCULATING PATH INTEGRALS

We have obtained the propagator of

METHODS OF CALCULATING PATH INTEGRALS 647

should talk about a technical problem that exists in Equation (20.80) In this expression, even though all the paths in C[ZO, to; X, t] are continuous, because

of the nature of the Brownian motion they zig zag The average distance squared covered by a Brown particle is given as

oc)

(2) = S _ _ m ( z , l ) 2 d x a t (20.83) From here we find the average distance covered during time t as

which gives the velocity of the particle at any point as

(20.86)

In summary: If we look a t the propagator [Eq (20.80)] as a probability distribution, it is Equation (20.79) written as a path integral, evaluated over all Brown paths with a suitable weight factor depending on the potential V(z) The zig zag motion of the particles in Brownian motion is essential in the fluid exchange process of living cells In fractal theory, paths of Brown particles are two-dimensional fractal curves The possible connections between fractals, path integrals, and differintegrals are active areas of research

20.6.1 Method of Time Slices

Let us evaluate the path integral of the functional F/z(T)] with the Wiener measure We slice a given path X(T) into N equal time intervals and approx- imate the path in each slice with a straight line I N ( T ) as

l,v(ti) = ti) = xi, i = 1,2,3, , N (20.87) This means that for a given path, z(T), and a small number E we can always find a number N = N ( E ) independent of T such that

648 GREEN'S FUNCTIONS A N D PATH INTEGRALS

Fig 20.3 Paths for the time slice method

is true Under these conditions for smooth functionals (Fig 20.3) the inequal- ity

is satisfied such that the limit lim,,o 6(~) i 0 is true Because all the infor- mation about E N ( T ) is contained in the set 2 1 = ~ ( t l ) , , z~ = z ( t ~ ) , we can also describe the functional F [ ~ N ( 7 ) ] by

which means that

(20.91)

METHODS OF CALCULATING PATH INTEGRALS 649

Because for N = 1, 2,3, , the function set FN(z~, 2 2 , , X N ) forms a Cauchy set approaching F [ X ( ~ ) ] , for a suitably chosen N we can use the integral

a Wiener path integral &jo,o;tl d,z(’r)F[x(7)] will be converted into an N-

dimensional integral [Eq (20.92)]

650 GREEN’S FUNCTIONS AND PATH INTEGRALS

From Equation (20.27), t,he value of the last integral is one Finally, using Equations (20.24) and (20.22), we obtain

(20.98)

In this calculation we have assumed that 7 lies in the last time slice denoted

by N + 1 Complicated functionals can be handled by Equation (20.92)

METHODS OF CALCULATING PATH INTEGRALS 651

Naturally, the major contribution to this integral comes from the paths that satisfy the Euler-Lagrange equation

(20.101)

We show these “classical” paths by 2 , ( 7 ) These paths also make the integral Ld7 an extremum, that is,

6 s L d r = 0 (20.102) However, we should also remember that in the Bloch equation V ( x ) is not quite the potential and L is not the Lagrangian Similarly, S~5d-r in Equa- tion (20.102) is not the action, S[Z(T)], of classical physics These expressions gain their conventional meanings only when we apply path integrals to the Schrdinger equation It is for this reason that we have used the term “semi- classical”

When the diffusion constant is much smaller than the functional S, that is,

D / S << 1, we write an approximate solution to Equation (20.100) as

where $(t - to) is called the fluctuation factor Even though methods

of finding the fluctuation factor are beyond our scope (see Chaichian and Demichev), we give two examples for its appearance and evaluation

Example 20.1 Evaluation of &zo,O;z,tl d,z(r): To find the propagator

W ( z , t, z0,O) we write

(20.104) and the Euler-Lagrange equation

2 J r ) = 0, Zc(O) = 2 0 , z(t) = z, (20.105) with the solution