Equations of motion Hamiltonian

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

This relationship is known as Liouville s equation. It is the fundamental theorem of statistical mechanics and is valid only for the phase space of generalized coordinates and momenta, whereas it is not, for example valid for the phase space of generalized coordinates and velocities. This is the basic reason that one chooses the Hamiltonian equations of motion in statistical mechanics. 

Here, Pp = mpR is a Cartesian bead momentum, U is the internal potential energy of the system of interest, Xi,.. .,Xk are a set of AT Lagrange multiplier constraint fields, which must be chosen so as to satisfy the K constraints, and is the rapidly fluctuating force exerted on bead p by interactions with surrounding solvent molecules. The corresponding Hamiltonian equation of motion is... 

We first construct an adiabatic approximation for the classical Hamiltonian equations of motion. We start from an unconstrained model with a classical... 

In the range of small noise intensities D, the optimal path q vi(t qf, 4>j) to the point (qf, f) is given by the condition that the action S be minimal. The variational problem for S to be extremal gives Hamiltonian equations of motion... 

At this stage the gauge of the vector potential is left free with the consequence that the evolution generated by the Hamiltonian equations of motion... 

The atomic motions change the configurations of the atoms (g f , gf), while the external forces, the forces between the atoms of the same molecule and the forces which act in every collision, change the velocities (and therefore also the momenta p, , p ). The corresponding changes in the phase of the gas model are expressed by the Hamiltonian equations of motion 69... 

Originally the equations of motion for the process were selected to be the Hamiltonian equations of motion obtained using the time-var5nng H. That is. 

Given the Lagrange s equations of motion (2.14) and the Hamiltonian function (2.22), the next task is to derive the Hamiltonian equations of motion for the system. This can be achieved by taking the differential of H defined by (2.22). Each side of the differential of H produces a differential expressed as ... 

These commutation rules together with the rules for converting the Hamiltonian equations of motion into matrix form constitute matrix mechanics, which is a way of stating the laws of quantum mechanics which is entirely different from that which we have used in this book, although completely equivalent. The latter rules require a discussion of differentiation with respect to a matrix, into which we shall not enter.1... 

The vectors and R +i/2 are vectors ofAfCO, 1) i.i.d. random numbers, with y > 0 the usual Langevin dynamics friction parameter. Setting y = 0 reduces the scheme to the usual RATTLE scheme for solving holonomically constrained Hamiltonian equations of motion, whereas if y is chosen large then this will be expected to cause instability in the scheme, as we are not solving the OU process exactly. The... 

The force and velocity are vectors, whose direction and magnitude are both of importance. In complex problems it is often preferable to reformulate classical mechanics in terms of a scalar, such as the energy, which is characterized only by its magnitude. This gives rise to the Lagrangian and Hamiltonian equations of motion. The latter equations are of most interest here and are dqi d K dt dpi ... 

These equations show that it is the classical action Sn that satisfies the Hamiltonian-Jacoby equation (3.11) with coordinate x, momentum p = dSn (x)/dx, and Hamiltonian equal to zero (stationary condition). The Hamiltonian equations of motion for the system are... 

Now we use the Hamiltonian equations of motion to obtain the ffuctua-tional trajectories ... 

For the system in exercise 9.1, determine the Hamiltonian equation of motion. 

A direct consequence of the equipartition theorem is the following using the hamiltonian equations of motion for such a system... 

This function is a constant of the motion and evaluates to Eg, the total energy of the extended system. The corresponding Hamiltonian equations of motion read... 

The coupled, first-order Hamiltonian equations of motion for the various systems studied were integrated numerically on either a CDC 7600 or a DEC VAX 11/780 digital computer using a variable step-size, fifth-order Adams-Moulton predictor-corrector integration technique As a test of accuracy, rate constants were computed for... 

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