ELSEVIER Physica A 245 (1997) 517-522

Stochastic dynamics of constrained systems M. Schwartz*, Y. Navot

Raymond an Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Received 1 April 1997

Abstract

We consider the stochastic dynamics of constrained systems. It is shown that by expressing the Lagrange multipliers in terms of quantities that are local in time, exact and potentially useful dynamic equations of the Langevin, Fokker-Planck and Smoluchowskii equations are obtained. The equations do not require the introduction of coordinates on the manifold to which the system is restricted but have a symmetric form and are described in terms of the coordinates of the embedding space.

1. Introduction

The dynamics of many interesting physical systems is complicated by constraints that

may be either global as in the spherical model or local as for an incompressible fluid. The formulation of equilibrium statistical systems seems straightforward but the con-

struction of dynamic stochastic equations to describe nonequilibrium statistical physics

is more tricky. The Fokker-Planck equation for the distribution function of constrained systems has been considered in the past [ 1-6]. The equation was obtained in terms of

the coordinates on the manifold to which the system is restricted and not derived from a Langevin equation in the embedding space. The purpose of the present paper is to obtain equations of motion that have a symmetric structure in the embedding space. We

start by constructing the equations of motion for a particle in d dimensions constrained to a ( d - 1)-dimensional manifold. (This is equivalent, of course, to a system of N particles in d ~ dimensions such that N d ~ = d , that have a single constraint on their coordinates.) We then go over to obtain the Langevin equation and the Fokker-Planck and Smoluchowskii equations following from it. (In this paper we adopt the name Fokker-Planck for the equation describing the evolution of the density in phase space

* Corresponding author.

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518 M. Schwartz, Y Navot/Physica A 245 (1997) 517-522

and Smoluchowskii for the equation describing the evolution in configuration space.)

The results are then generalized to more than one constraint.

2. The equations of motion for a single constraint

Consider a particle moving in d dimensions under the influence of a potential W

and constrained to a surface ~k(r) = 0. The particle obeys the equation

mi: = - • W + 2n , (1)

where m is the mass of the particle, n = V~,/IV~b I is the unit vector normal to

the hypersurface to which the particle is restricted and 2 is the appropriate Lagrange

multiplier, ensuring, that for all times ~k(r ( t ) ) = O.

Now, Eq. (1) is not very useful for our purpose. (This is true also if a numerical

solution is attempted.) The reason is that the equation is not really local in time. To

solve it, first, it has to be solved for general 2(0. The solution is then r = r{2, t}, a

functional of 2 and a function of t and then 2 (0 has to be obtained from the condition ~k[r{2, t}] = 0. This can be circumvented by actually obtaining 2 in terms of quantities

that are local in time. Since the motion of the particle is constrained to the surface ~b(r) = 0, we must

have

e - n = 0 . ( 2 )

Taking the time derivative of Eq. (2) and substituting r from Eq. (1) we obtain

2 = V W . n - m0: ' C i ) , (3)

where C is the curvature tensor

, ~3n: :¢ - nknj ) ] Cij = (t~i: - n in : )~xk t, ok j

The equation of motion reads now

mi: = - P ~ T W - mi" . (C~)n, (4)

where P is the matrix projecting locally on the surface (P i j = ~ i j - n i n j ) . The advantage of the form (4) over (1) is that, now in order to evolve r in time we need only local

quantities and not some unknown function 2 (0 to be adjusted in the end. The first term on the right-hand side is tangential to the surface while the second is

normal. To understand the significance of the second term choose locally the directions of the coordinate axes as the directions of the principal axes of the curvature tensor. The curvature tensor is diagonalized and along the diagonal we find [ l /R1 . . . . . 1 /Rd_ l ,O]

where R1, . . . , R d - I are the principal radii of curvature. I f we decompose r along those

M. Schwartz, Y. NavotlPhysica A 245 (1997) 517-522 519

axes we find that the normal force, fn, is

( d - l ) ~:2

~-n. (5) f n = - m Z Ri i=1

This is just the centripetal force needed to keep the particle on the surface. It is obvious that Eq. (3) has to be supplemented by initial conditions r(0) and i(0) such that ~O(r(0)) = 0 and i (0 ) . n(r(O)) = O.

3. Fokker-Planck and Smluchowskii equations

The Langevin equation is obtained from Eq. (3) by adding to it a friction term, - ~ = -~PF and a noise term Pg(t), where the average of the noise is zero and the correlation of two Cartesian components of the noise are given by

(rh(t)11j(fl)) = 2D6q~(t - t') . (6)

The calculation of 2 does not change, since it results only from the constraint. We now write down the Langevin equation as two equations which enable us to write down the Fokker-Planck equation at once.

u = i (7)

and

il = - P VW + - u + u (Cu)n m mr/ - • . (8)

The Fokker-Planck equation for P(r,u), the probability density to find the particle at r having a velocity u, is

= V u . P Vu+ V W + m u +(u . Cu)n P+Vr" [-PUP]. (9)

It is clear that P(r, u, 0), the initial condition for Eq. (9), must vanish for all r and u that do not obey the conditions ~(r) = 0 and u. n(r) = 0. Since the equations preserve these conditions at any later time we can define a function Q(r, u, t) by the relation

P(r,u,t) = Q(r,u,t)6(~k(r)) l V~16(u . n(r)). (10)

(Note that 6(~O)]V~b] is invariant under the parametrization of the same surface. Namely, if we replace q/(r) by ~b(r) such that the surfaces ~,(r) = 0 and ~b(r) = 0 are identical then 6(~b)IV~k I = 6(tp)lV~b[.)

Some algebra is needed for obtaining an equation for Q,

O--~ = - ~ + VrW+mU +(u . Cu)n. V u - P u . Vr Q. (11)

520 M. Schwartz, Y. Navot/Physica A 245 (1997) 517-522

Note that Eq. (11) is not unique because actually both sides of the equations are multiplied by 6(u. n)6(~)1~7~1 (from the left) and so we can add to the right-hand side of Eq. (11) any term that vanishes under multiplication by 6(u. n)lV¢,l~(~).

It may now be easily verified that the equilibrium solution of Eq. (11) is,

Qcx~exp - ~ m f f f +

where kT = a/D. (To prove that we have to carry in mind again that Eq. ( 11 ) is actually multiplied by

6(u.n)6(~,)lV~l. For example, it seems that the term (u. Cu)n. ~TuQ violates the above solution but it produces a term (u. Cu)(m)/kT)(n . u) that vanishes when multiplied by 3(u. n).) The equilibrium distribution function is given thus by

[ /:I .2 W(r)X~] Peq(r,u) O( ccexp - ~sm-ff-~ + kT JJ 3(ff)lV~klf(u" n) , (12)

as may have been expected. The time-dependent solution, though, is, of course, not the product of the unconstrained solution into the constraining 6(O)]V~lf(u. n).

Next we derive the corresponding Smoluchowskii equation by integration of the equation for P(r, u) over u. (An alternative approach would be to consider the Langevin equation and approximate it by dropping the acceleration terms as the basis for con- structing the equation for the distribution P(r, t) of finding the particle in the vicinity of r at time t.)

Denoting P ( r ) - - f d a u P ( r , u ) , we obtain the equation of continuity (from Eq. (9))

- - V . J , ~t

where J, the probability current density in configuration space, is given by

s(.) = f adue.P(.,.)= f

(13)

(14)

To obtain an equation for J we multiply both sides of Eq. (9) by u and integrate over u. We obtain

8--~J=P[-l~TwP]-~J+/ddu(u'Cu)nP(r'u)-/dduuPu'~rP(r'u)'St m

(15)

(Note that multiplication by Pu would not have given on the left-hand side of Eq. (15) (~J/~t) but rather P(~J/~t).)

Eq. (15) is exact. To close it we have to approximate. We consider first the compo- nent projected on the surface, separately. Then we assume that the velocities equilibrate faster than the coordinates so that in the last term on the right-hand side we can use the equilibrium distribution of the velocity and then we drop the tangential acceleration term P(~J/~t). We obtain that the current density is given by

J(r) = - 1 p [ v w + kTV]P. (16) o~

M. Schwartz, Y. NavotlPhysica A 245 (1997) 517-522 521

So, the equation of continuity has the form

0/5 I V . {P[VW + kVV]/5}. (17) c~t

It is clear that/5 has the form O6(qJ)lV~bl and it is easily verified that 0 obeys the

following equation:

~0 1 - P V . { P [ V W + k T V ] O } , (18)

at

and as expected the equilibrium solution following from Eq. (18) is

Qcx exp I - ~--~) ] 6(~b)(xT~), (19)

that is consistent, as it should, with the result obtained by integrating Peq(r, u) over u.

4. More than one constraint

Suppose now that the particle is constrained to a manifold of lower dimension defined by {qJi(r) = 0}. The equation of motion (in the absence of friction and noise) will

now read

mk" = - ~ W + Z 2 i n i ( r ) ' (20)

where ni(r) is the unit vector perpendicular to ~i(r) at a point on the surface. By differentiating ~. n i ( r ) ---- 0 , it is found that the Lagrange multipliers 2i are obtained by solving the following linear system:

Aij,~ j : - mi' . c ( i ) i • + V W • nz , (21)

where the matrix elements Aij are n i . n j and C ~) is the curvature tensor corresponding

to the surface ~bi(r) = O.

Once the 2's are obtained the procedure follows the same steps as for a single constraint. Of particular importance, are systems where independent sets of coordinates are constrained. To give a very common example. In On models, we consider interacting rotators of unit length. Such a system can be described in terms of n fields {~b j} at each point i such that n J 2 ~--]j=l(tpl) = 1. In such a case the matrix A is the unit matrix

and the basic equation of motion is

mi" = - P V W - m~. C r Y , (22)

where P projects on the manifold to which the particle is restricted and Cr, the total curvature tensor is the sum of the C(i)s. In fact P is the product of P,-'s, each projecting on one of the surfaces, while Cr is a matrix with non zero blocks C (i) along the diagonal.

We expect to publish in the very near future a work on the dynamics of the spherical random field model, based on the above formulation.

522 M. Schwartz, Y. Naoot/Physica A 245 (1997) 517-522

Acknowledgements

This work was supported by a grant from the Israel Science Foundation.

References

[1] N.G. van Kampen, J. Statist. Phys. 44 (1986) 1. [2] H. Risken, The Fokker Planck Equation, Springer, Berlin, 1989. [3] K. Ito, Nagoya Math. J. 1 (1950) 35. [4] H.P. McKean Jr., Stochastic Integrals, Academic Press, New York, 1969. [5] K.D. Elworthy, Stochastic Differential Equations on Manifolds, London Mathematical Society Lecture

Note Series, vol. 70, Cambridge University Press, Cambridge, 1982. [6] M. van den Berg, Y. Lewis, Bull. Lond. Math. Soc. 17 (1985) 144.