Path integral molecular dynamics (optional reading) next up previous
Next: Path integrals for N-particle Up: No Title Previous: Thermodynamics from path integrals

Path integral molecular dynamics (optional reading)

Consider once again the path integral expression for the one-dimensional canonical partition function (for a finite but large value of P):

  equation253

(the condition tex2html_wrap_inline522 is understood). Recall that, according to the classical isomorphism, the path integral expression for the canonical partition function is isomorphic to the classical configuration integral for a certain P-particle system. We can carry this analogy one step further by introducing into the above expression a set of P momentum integrations:

  equation263

Note that these momentum integrations are completely uncoupled from the position integrations, and if we were to carry out these momentum integrations, we would reproduce Eq. (1) apart from trivial constants. Written in the form Eq. (2), however, the path integral looks exactly like a phase space integral for a P-particle system. We know from our work in classical statistical mechanics that dynamical equations of motion can be constructed that will generate this partition function. In principle, one would start with the classical Hamiltonian

displaymath273

derive the corresponding classical equations of motion and then couple in thermostats. Such an approach has certainly been attempted with only limited success. The difficulty with this straightforward approach is that the more ``quantum'' a system is, the large the paramester P must be chosen in order to converge the path integral. However, if P is large, the above Hamiltonian describes a system with extremely stiff nearest-neighbor harmonic bonds interacting with a very weak potential U/P. It is, therefore, almost impossible for the system to deviate far harmonic oscillator solutions and explore the entire available phase space. The use of thermostats can help this problem, however, it is also exacerbated by the fact that all the harmonic interactions are coupled, leading to a wide variety of time scales associated with the motion of each variable in the Hamiltonian. In order to separate out all these time scales, one must somehow diagonalize this harmonic interaction. One way to do this is to use normal mode variables, and this is a perfectly valid approach. However, we will explore another, simpler approach here. It involves the use of a variable transformation of the formed used in previous lectures to do the path integral for the free-particle density matrix.

Consider a change of variables:

eqnarray280

where

displaymath283

The inverse of this transformation can be worked out in closed form:

eqnarray287

and can also be expressed as a recursive inverse:

eqnarray292

The term k=P here can be used to start the recursion. We have already seen that this transformation diagonalized the harmonic interaction. Thus, substituting the transformation into the path integral gives:

displaymath297

The parameters tex2html_wrap_inline538 are given by

eqnarray303

Note also that the momentum integrations have been changed slightly to involve a set of parameters tex2html_wrap_inline540 . Introducing these parameters, again, only changes the partition function by trivial constant factors. How these should be chosen will become clear later in the discussion. The notation tex2html_wrap_inline542 indicates that each variable tex2html_wrap_inline544 is a generally a function of all the new variables tex2html_wrap_inline546 .

A dynamics scheme can now be derived using as an effective Hamiltonian:

displaymath306

which, when coupled to thermostats, yields a set of equations of motion

eqnarray312

These equations have a conserved energy (which is not a Hamiltonian):

displaymath322

Notice that each variable is given its own thermostat. This is done to produce maximum ergodicity in the trajectories. In fact, in practice, the chain thermostats you have used in the computer labs are employed. Notice also that the time scale of each variable is now clear. It is just determined by the parameters tex2html_wrap_inline548 . Since the object of using such dynamical equations is not to produce real dynamics but to sample the phase space, we would really like each variable to move on the same time scale, so that there are no slow beads trailing behind the fast ones. This effect can be produced by choosing each parameter tex2html_wrap_inline540 to be proportional to tex2html_wrap_inline538 : tex2html_wrap_inline554 . Finally, the forces on the u variables can be determined easily from the chain rule and the recursive inverse given above. The result is

eqnarray334

where the first (i=1) of these expressions starts the recursion in the second equation.

Later on, when we discuss applications of path integrals, we will see why a formulation such as this for evaluating path integrals is advantageous.


next up previous
Next: Path integrals for N-particle Up: No Title Previous: Thermodynamics from path integrals

Mark Tuckerman
Mon Mar 29 18:17:04 EST 1999