The Microscopic Connection

Classical mechanics (Goldstein, 1980) is based on Newton's three laws of motion. This theory introduced the concepts of a force and an acceleration. Prior to Newton's work, the connection had been made between forces and velocities. Newton's laws of motion were supplemented by the notion of a force acting at a distance. With the identification of the force of gravity and an appropriate initial condition - initial coordinates and velocities - trajectories could be computed. Philosophers of science have debated the content of Newton's laws but when augmented with a force which is expressible as a function of time, position or possibly of velocity, those laws lead to the equation,

(3.1)

which is well-posed and possesses a unique solution.

After Newton, scientists discovered different sets of equivalent laws or axioms upon which classical mechanics could be based. More elegant formulations are due to Lagrange and Hamilton. Newton's laws are less general than they might seem. For instance the position r, that appears in Newton's equation must be a Cartesian vector in a Euclidean space. One does not have the freedom of say, using angles as measures of position. Lagrange solved the problem of formulating the laws of mechanics in a form which is valid for generalised coordinates.

Let us consider a system with generalisedcoordinates q. These coordinates may be Cartesian positions, angles or any other convenient parameters that can be found to uniquely specify the configuration of the system. The kinetic energy T, will in general be a function of the coordinates and their time derivatives . If V(q) is the potential energy, we define the Lagrangian to be . The fundamental dynamical postulate states that the motion of a system is such that the action, S, is an extremum

(3.2)

Let q(t) be the coordinate trajectory that satisfies this condition and let q(t)+δq(t) where δq(t) is an arbitrary variation in q(t), be an arbitrary trajectory. The varied motion must be consistent with the initial and final positions. So that, δq(t₁)=δq(t₀)=0. We consider the change in the action due to this variation.

(3.3)

Integrating the second term by parts gives

(3.4)

The first term vanishes because δq is zero at both limits. Since for t₀<t<t₁, δq(t) is arbitrary, the only way that the variation in the action δS, can vanish is if the equation,

(3.5)

holds for all time. This is Lagrange's equation of motion. If the coordinates are taken to be Cartesian, it is easy to see that Lagrange's equation reduces to Newton's.

Although Lagrange's equation has removed the special status attached to Cartesian coordinates, it has introduced a new difficulty. The Lagrangian is a function of generalised coordinates, their time derivatives and possibly of time. The equation is not symmetric with respect to the interchange of coordinates and velocities. Hamilton derived an equivalent set of equations in which the roles played by coordinates and velocities can be interchanged. Hamilton defined the canonical momentum p,

(3.6)

and introduced the function

(3.7)

This function is of course now known as the Hamiltonian. Consider a change in the Hamiltonian which can be written as

(3.8)

The Lagrangian is a function of q, and t so that the change dL, can be written as

(3.9)

Using the definition of the canonical momentum p, and substituting for dL, the expression for dH becomes

(3.10)

Lagrange's equation of motion (3.5), rewritten in terms of the canonical momenta is

(3.11)

so that the change in H is

(3.12)

Since the Hamiltonian is a function of q,p and t, it is easy to see that Hamilton equations of motion are

and (3.13)

As mentioned above these equations are symmetric with respect to coordinates and momenta. Each has equal status in Hamilton's equations of motion. If H has no explicit time dependence, its value is a constant of the motion. Other formulations of classical mechanics such as the Hamilton-Jacobi equations will not concern us in this book.

Apart from relativistic or quantum corrections, classical mechanics is thought to give an exact description of motion. In this section our point of view will change somewhat. Newtonian or Hamiltonian mechanics imply a certain set of constants of the motion: energy, and linear and angular momentum. In thermodynamically interesting systems the natural fixed quantities are the thermodynamic state variables; the number of molecules N, the volume V and the temperature T. Often the pressure rather than the volume may be preferred. Thermodynamically interesting systems usually exchange energy, momentum and mass with their surroundings. This means that within thermodynamic systems none of the classical constants of the motion are actually constant.

Typical thermodynamic systems are characterised by fixed values of thermodynamic variables: temperature, pressure, chemical potential, density, enthalpy or internal energy. The system is maintained at a fixed thermodynamic state (say temperature) by placing it in contact with a reservoir, with which it exchanges energy (heat) in such a manner as to keep the temperature of the system of interest fixed. The heat capacity of the reservoir must be much larger than that of the system, so that the heat exchanged from the reservoir does not affect the reservoir temperature.

Classical mechanics is an awkward vehicle for describing this type of system. The only way that thermodynamic systems can be treated in Newtonian or Hamiltonian mechanics is by explicitly modelling the system, the reservoir and the exchange processes. This is complex, tedious and as we will see below, it is also unnecessary. We will now describe a little known principle of classical mechanics which is extremely useful for designing equations of motion which are more useful from a macroscopic or thermodynamic viewpoint. This principle does indeed allow us to modify classical mechanics so that thermodynamic variables may be made constants of the motion.

Just over 150 years ago Gauss formulated a mechanics more general than Newton's. This mechanics has as its foundation Gauss' principle of least constraint. Gauss (1829) referred to this as the most fundamental dynamical principle (Whittacker 1937, Pars 1965). Suppose that the cartesian coordinates and velocities of a system are given at time t. Consider the function C, referred to by Hertz as the square of the curvature, where

(3.14)

C is a function of the set of accelerations . Gauss' principle states that the actual physical acceleration corresponds to the minimum value of C. Clearly if the system is not subject to a constraint then C=0 and the system evolves under Newton's equations of motion. For a constrained system it is convenient to change variables from r_i to w_i where

(3.15)

Because the {w_i}, are related to the Jacobi metric, we will refer to this coordinate system as the Jacobi frame.

The types of constraints which might be applied to a system fall naturally into two types, holonomic and nonholonomic. A holonomic constraint is one which can be integrated out of the equations of motion. For instance, if a certain generalised coordinate is fixed, its conjugate momentum is zero for all time, so we can simply consider the problem in the reduced set of unconstrained variables. We need not be conscious of the fact that a force of constraint is acting upon the system to fix the coordinate and the momentum. An analysis of the two dimensional motion of an ice skater need not refer to the fact that the gravitational force is exactly resisted by the stress on the ice surface fixing the vertical coordinate and velocity of the ice skater. We can ignore these degrees of freedom.

Nonholonomic constraints usually involve velocities. These constraints are not integrable. In general a nonholonomic constraint will do work on a system. Thermodynamic constraints are invariably nonholonomic. It is known that the Action Principle cannot be used to describe motion under nonholonomic constraints (Evans and Morriss, 1984).

We can write a general constraint in the Jacobi frame in the form

(3.16)

where g is a function of Jacobi positions, velocities and possibly time. Either type of constraint function, holonomic or nonholonomic, can be written in this form. If this equation is differentiated with respect to time, once for nonholonomic constraints and twice for holonomic constraints we see that,

(3.17)

We refer to this equation as the differential constraint equation and it plays a fundamental role in Gauss' Principle of Least Constraint. It is the equation for a plane which we refer to as the constraint plane. n is the vector normal to the constraint plane.

Our problem is to solve Newton's equation subject to the constraint. Newton's equation gives us the acceleration in terms of the unconstrained forces. The differential constraint equation places a condition on the acceleration vector for the system. The differential constraint equation says that the constrained acceleration vector must terminate on a hyper-plane in the 3N-dimensional Jacobi acceleration space (equation 3.17).

Imagine for the moment that at some initial time the system satisfies the constraint equation g=0. In the absence of the constraint the system would evolve according to Newton's equations of motion where the acceleration is given by

(3.18)

This trajectory would in general not satisfy the constraint. Further, the constraint function g tells us that the only accelerations which do continuously satisfy the constraint, are those which terminate on the constraint plane. To obtain the constrained acceleration we must project the unconstrained acceleration back into the constraint plane.

Gauss' principle of least constraint gives us a prescription for constructing this projection. Gauss' principle states that the trajectories actually followed are those which deviate as little as possible, in a least squares sense, from the unconstrained Newtonian trajectories. The projection which the system actually follows is the one which minimises the magnitude of the Jacobi frame constraint force. This means that the force of constraint must be parallel to the normal of the constraint surface. The Gaussian equations of motion are then

(3.19)

where λ is a Gaussian multiplier which is a function of position, velocity and time.

To calculate the multiplier we use the differential form of the constraint function. Substituting for the acceleration we obtain

(3.20)

It is worthwhile at this stage to make a few comments about the procedure outlined above. First, notice that the original constraint equation is never used explicitly. Gauss' principle only refers to the differential form of the constraint equation. This means that the precise value of the constrained quantity is undetermined. The constraint acts only to stop its value changing. In the holonomic case Gauss' principle and the principle of least action are of course completely equivalent. In the nonholonomic case the equations resulting from the application of Gauss' Principle cannot be derived from a Hamiltonian and the principle of least action cannot be used to derive constraint satisfying equations. In the nonholonomic case, Gauss' principle does not yield equations of motion for which the work done by the constraint forces is a minimum.

The derivation of constrained equations of motion given above is geometric, and is done in the transformed coordinates which we have termed the Jacobi frame. It is not always convenient to write a constraint function in the Jacobi frame, and from an operational point of view a much simpler derivation of constrained equations of motion is possible using Lagrange multipliers. The square of the curvature C is a function of accelerations only (the Cartesian coordinates and velocities are considered to be given parameters). Gauss' principle reduces to finding the minimum of C, subject to the constraint. The constraint function must also be written as a function of accelerations, but this is easily achieved by differentiating with respect to time. If G is the acceleration dependent form of the constraint, then the constrained equations of motion are obtained from

(3.21)

It is easy to see that the Lagrange multiplier λ, is (apart from the sign) equal to the Gaussian multiplier. We will illustrate Gauss' principle by considering some useful examples.

The most common type of holonomic constraint in statistical mechanics is probably that of fixing bond lengths and bond angles in molecular systems. The vibrational degrees of freedom typically have a relaxation timescale which is orders of magnitude faster than translational degrees of freedom, and are therefore often irrelevant to the processes under study. As an example of the application of Gauss' principle of least constraint for holonomic constraints we consider a diatomic molecule with a fixed bond length. The generalisation of this method to more than one bond length is straightforward (see Edberg, Evans and Morriss, 1986) and the application to bond angles is trivial since they can be formulated as second nearest neighbour distance constraints. The constraint function for a diatomic molecule is that the distance between sites one and two be equal to d₁₂, that is

, (3.22)

where we define r₁₂ to be the vector from r₁ to r₂, (r₁₂ ≡ r₂-r₁). Differentiating twice with respect to time gives the acceleration dependent constraint equation,

(3.23)

To obtain the constrained equations of motion we minimise the function C subject to the constraint equation (3.23). That is

(3.24)

For i equal to 1 and 2 this gives

(3.25)

Notice that the extra terms in these equations of motion have opposite signs. This is because the coefficients of the r₁ and r₂ accelerations have opposite signs. The total constraint force on the molecule is zero so there is no change in the total momentum of the molecule. To obtain an expression for the multiplier λ we combine these two equations to give an equation of motion for the bond vector r₁₂,

(3.26)

Substituting this into the differential form of the constraint function (3.23), gives

(3.27)

It is very easy to implement these constrained equations of motion as the multiplier is a simple explicit function of the positions, velocities and Newtonian forces. For more complicated systems with multiple bond length and bond angle constraints (all written as distance constraints) we obtain a set of coupled linear equations to solve for the multipliers.

One of the simplest and most useful applications of Gauss' Principle is to derive equations of motion for which the ideal gas temperature (ie. the kinetic energy) is a constant of the motion (Evans et. al. 1983). Here the constraint function is

(3.28)

Differentiating once with respect to time gives the equation for the constraint plane

(3.29)

Therefore to obtain the constrained Gaussian equations we minimise C subject to the constraint equation (3.29). That is

(3.30)

This gives

(3.31)

Substituting the equations of motion into the differential form of the constraint equation, we find that the multiplier is given by

(3.32)

As before, λ is a simple function of the forces and velocities so that the implementation of the constant kinetic energy constraint in a molecular dynamics computer program only requires a trivial modification of the equations of motion in a standard program. Equations (3.31 & 32) constitute what have become known as the Gaussian isokinetic equations of motion. These equations were first proposed simultaneously and independently by Hoover et. al. (1982) and Evans (1983). In these original papers Gauss' principle was however not referred to. It was a year before the connection with Gauss' principle was made.

With regard to the general application of Gauss' principle of least constraint one should always examine the statistical mechanical properties of the resulting dynamics. If one applies Gauss' principle to the problem of maintaining a constant heat flow, then a comparison with linear response theory shows that the Gaussian equations of motion cannot be used to calculate thermal conductivity (Hoover 1986). The correct application of Gauss' principle is limited to arbitrary holonomic constraints and apparently, to nonholonomic constraint functions which are homogeneous functions of the momenta.