For many systems it is apparent that after possible initial transients lasting a time t0, the N particle distribution function f(Γ,t), becomes essentially time independent. This is evidenced by the fact that the macroscopic properties of the system relax to fixed average values. This obviously happens for equilibrium systems. It also occurs in some nonequilibrium systems, so-called nonequilibrium steady states. We will call all such systems stationary.
For a stationary system, we may define the ensemble average of a phase variable B(Γ), using the stationary distribution function f(Γ), so that
(3.58)
On the other hand we may define a time average of the same phase variable as,
(3.59)
where t0 is the relaxation time required for the establishment of the stationary state. An ergodic system is a stationary system for which the ensemble and time averages of usual phase variables, exist and are equal. By usual we mean phase variable representations of the common macroscopic thermodynamic variables (see §3.7).
It is commonly believed that all realistic nonlinear many body systems are ergodic.
Example
We can give a simple example of ergodic flow if we take the energy surface to be the two-dimensional unit square 0<p<1 and 0<q<1. We shall assume that the equations of motion are given by
(3.60)
and we impose periodic boundary conditions on the system. These equations of motion can be solved to give
(3.61)
The phase space trajectory on the energy surface is given by eliminating t from these two equations
(3.62)
If α is a rational number, α=m∕n, then the trajectory will be periodic and will repeat after a period T=n. If α is irrational, then the trajectory will be dense on the unit square but will not fill it. When α is irrational the system is ergodic. To show this explicitly consider the Fourier series expansion of an arbitrary phase function A(q,p),
(3.63)
We wish to show that the time average and phase average of A(q,p) are equal for α irrational. The time average is given by
(3.64)
For irrational α, the denominator can never be equal to zero, therefore
(3.65)
Similarly we can show that the phase space average of A is
(3.66)
and hence the system is ergodic. For rational α the denominator in (3.64) does become singular for a particular jk-mode. The system is in the pure state labelled by jk. There is no mixing.
Ergodicity does not guarantee the relaxation of a system toward a stationary state. Consider a probability density which is not constant over the unit square, for example let f(q,p,t=0) be given by
(3.67)
then at time t, under the above dynamics (with irrational α), it will be
(3.68)
The probability distribution is not changed in shape, it is only displaced. It has also not relaxed to a time independent equilibrium distribution function. However after an infinite length of time it will have wandered uniformly over the entire energy surface. It is therefore ergodic but it is termed non mixing.
It is often easier to show that a system is not ergodic, rather than to show that it is ergodic. For example the phase space of a system must be metrically transitive for it to be ergodic. That is, all of phase space, except possibly a set of measure zero, must be accessible to almost all the trajectories of the system. The reference to almost all, is because of the possibility that a set of initial starting states of measure zero, may remain forever within a subspace of phase space which is itself of measure zero. Ignoring the more pathological cases, if it is possible to divide phase space into two (or more) finite regions of nonzero measure, so that trajectories initially in a particular region remain there forever, then the system is not ergodic. A typical example would be a system in which a particle was trapped in a certain region of configuration space. Later we shall see examples of this specific type.
If we consider two harmonic oscillators (see §3.2) which have the same frequency ω but different initial conditions (x1,p1) and (x2,p2), we can define the distance between the two phase points by
(3.69)
Using the equation for the trajectory of the harmonic oscillator (3.34), we see that as a function of time this distance is given by
(3.70)
where xi(t) and pi(t) are the position and momenta of oscillator i, at time t. This means that the trajectories of two independent harmonic oscillators always remain the same distance apart in Γ-space.
This is not the typical behaviour of nonlinear systems. The neighbouring trajectories of most N-body nonlinear systems tend to drift apart with time. Indeed it is clear that if a system is to be mixing then the separation of neighbouring trajectories is a precondition. Weakly coupled harmonic oscillators are an exceptions to the generally observed trajectory separation. This was a cause of some concern in the earliest dynamical simulations (Fermi, Pasta & Ulam, 1955).
As the separation between neighbouring trajectories can be easily calculated in a classical mechanical simulation, this has been used to obtain quantitative measures of the mixing properties of nonlinear many-body systems. If we consider two N-body systems composed of particles which interact via identical sets of interparticle forces, but whose initial conditions differ by a small amount, then the phase space separation is observed change exponentially as
(3.71)
At intermediate times the exponential growth of d(t) will be dominated by the fastest growing direction in phase space (which in general will change continuously with time). This equation defines the largest Lyapunov exponent λ for the system (by convention λ is defined to be real, so any oscillating part of the trajectory separation is ignored). For the harmonic oscillator the phase separation is a constant of the motion and therefore the Lyapunov exponent λ, is zero. In practical applications this exponential separation for an N particle system continues until it approaches a limit imposed by the externally imposed boundary conditions - the container walls, or the energy, or other thermodynamic constraints on the system (§ 7.8). If the system has energy as a constant of the motion then the maximum separation is the maximum distance between two points on the energy hypersphere. This depends upon the value of the energy and the dimension of the phase space.
The largest Lyapunov exponent indicates the rate of growth of trajectory separation in phase space. If we consider a third phase point Γ3(t), which is constrained such that the vector between Γ1 and Γ3 is always orthogonal to the vector between Γ1 and Γ2, then we can follow the rate of change of a two dimensional area in phase space. We can use these two directions to define an area element V1(t), and rate of change of the volume element is given by
(3.72)
As we already know the value of λ1, this defines the second largest Lyapunov exponent λ2. In a similar way, if we construct a third phase space vector Γ14(t) which is constrained to be orthogonal to both Γ12(t) and Γ13(t), then we can follow the rate of change of a three dimensional volume element V3(t) and calculate the third largest exponent λ3;
(3.73)
This construction can be generalised to calculate the full spectrum of Lyapunov exponents for an N particle system. We consider the trajectory Γ(t) of a dynamical system in phase space and study the convergence or divergence of neighbouring trajectories by taking a set of basis vectors (tangent vectors) in phase space {δ1,δ2,δ3,...}, where δi=Γi=Γ0. Some care must be exercised in forming the set of basis vectors to ensure that the full dimension of phase space is spanned by the basis set, and that the basis set is minimal. This simply means that constants of the motion must be considered when calculating the dimension of accessible phase space. If the equation of motion for a trajectory is of the form
(3.74)
then the equation of motion for the tangent vector δi is
(3.75)
Here T(Γ) is the Jacobian matrix (or stability matrix ∂G∕∂Γ)for the system. If the magnitude of the tangent vector is small enough the nonlinear terms in equation (3.75) can be neglected. The formal solution of this equation is
(3.76)
The mean exponential rate of growth of the ith tangent vector, gives the ith Lyapunov exponent
(3.77)
The existence of the limit is ensured by the multiplicative ergodic theorem of Oseledec [1968] (see also Eckmann and Ruelle [1985]). The Lyapunov exponents can be ordered λ1>λ2>...>λM and if the system is ergodic, the exponents are independent of the initial phase Γ(0) and the initial phase space separation δi(0).
If we consider the volume element VN where N is the dimension of phase space then we can show that the phase space compression factor gives the rate of change of phase space volume, and that this is simply related to the sum of the Lyapunov exponents by
(3.78)
For a Hamiltonian system, the phase space compression factor is identically zero, so the phase space volume is conserved. This is a simple consequence of Liouville's theorem. From equation (3.78) it follows that the sum of the Lyapunov exponents is also equal to zero. If the system is time reversible then the Lyapunov exponents occur in pairs (-λi,λi). This ensures that d(t), V2(t), V3(t), etc. change at the same rate with both forward and backward time evolution. It is generally believed that it is necessary to have at least one positive Lyapunov exponent for the system to be mixing. In chapters 7 and 10 we will return to consider Lyapunov exponents in both equilibrium and nonequilibrium systems.