Hamiltonian dynamics

Fet us consider a system described by an explicitly time-dependent Hamiltonian p, q.t) where (p, q) = z is a point in phase space. Hamilton s equation of motion are 

The probability density p(p, q, t) in phase space satisfies the continuity equation 

From Hamilton s equations, we find that dp/dp = —dq/dq, such that 

This is Liouville s equation, with the Liouville operator 

Equation (5.19) states that the flow in phase space is incompressible. In particular, following along a trajectory starting in (p(0), q(0)), we find that 

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F. Kang. The Hamiltonian way for computing Hamiltonian dynamics. In R. Spigler, editor. Applied and Industrial Mathematics, pages 17-35. Kluwer Academic, Dordrecht, The Netherlands, 1990. 

We note that for the harmonic Hamiltonian in (5.25) the variance of the work approaches zero in the limit of an infinitely slow transformation, v — 0, r — oo, vt = const. However, as shown by Oberhofer et al. [13], this is not the case in general. As a consequence of adiabatic invariants of Hamiltonian dynamics, even infinitely slow transformations can result in a non-delta-like distribution of the work. Analytically solvable examples for that unexpected behavior are, for instance, harmonic Hamiltonians with time-dependent spring constants k = k t). 

This procedure follows, in effect, the derivation of Jarzynski s identity in discrete time [2,18], as outlined in Sect. 5.5. Finally, for Hamiltonian dynamics, one can use (5.23) and calculate the work directly from the difference in total energy between trajectory start and end points. 

For Hamiltonian dynamics with a canonical or microcanonical distribution of initial conditions the acceptance probability for pathways generated with the shifting algorithm is particularly simple. Provided forward and backward shifting moves are carried out with the same probability the acceptance probability from (7.23) reduces to... 

For the purposes of the present treatment, we wish to rewrite this trajectory average as an average over the initial, equilibrium distribution. If the system evolves according to deterministic (e.g., Hamiltonian) dynamics, each trajectory is uniquely determined by its initial point, and (8.46) can be written without modification as an average over the canonical phase space distribution. 

The classical spin motion that follows from the above considerations can be viewed as a dynamics on the sphere S2 driven by the Hamiltonian dynamics in phase space. To see this one first transforms the (redundant) fourdimensional representation of the matrix degrees of freedom, corresponding... 

This equation describes the Thomas precession of a classical spin (Thomas, 1927), which is driven by the underlying Hamiltonian dynamics in phase space. In combination the two types of dynamics, Hamiltonian on phase space and driven precession on the sphere, yield the following picture There is a combined phase space It3 x It3 x S2 with two dynamical systems,... 

MSN.196. 1. Prigogine, S. Kim, G. Ordonez, and T. Petrosky, Stochasticity and time symmetry breaking in Hamiltonian dynamics, in Proceedings, XXII Solvay Conference Physics, The Physics of Communication, Delphi, 2001, World Scientific, Singapore, 2003, pp. 1-22. 

A major preoccupation of nonequilibrium statistical mechanics is to justify the existence of the hydrodynamic modes from the microscopic Hamiltonian dynamics. Boltzmann equation is based on approximations valid for dilute fluids such as the Stosszahlansatz. In the context of Boltzmann s theory, the concept of hydrodynamic modes has a limited validity because of this approximation. We may wonder if they can be justified directly from the microscopic dynamics without any approximation. If this were the case, this would be great progress... 

A major preoccupation in nonequilibrium statistical mechanics is to derive hydrodynamics and nonequilibrium thermodynamics from the microscopic Hamiltonian dynamics of the particles composing matter. The positions raYl= and momenta PaY i= of these particles obey Newton s equations or, equivalently, Hamilton s equations ... 

The time-reversal symmetry of the Hamiltonian dynamics, also called the microreversibility, is the property that if the phase-space trajectory... 

The time evolution of the probability density is induced by Hamiltonian dynamics so that it has its properties—in particular, the time-reversal symmetry. However, the solutions of Liouville s equation can also break this symmetry as it is the case for Newton s equations. This is the case if each trajectory (43) has a different probability weight than its time reversal (44) and that both are physically distinct (45). 

The singular character of the diffusive modes allows their exponential relaxation at the rate given by the dispersion relation of diffusion. Their explicit construction can be used to perform an ab initio derivation of entropy production directly from the microscopic Hamiltonian dynamics [8, 9]. 

Further large-deviation dynamical relationships are the so-called flucmation theorems, which concern the probability than some observable such as the work performed on the system would take positive or negative values under the effect of the nonequilibrium fluctuations. Since the early work of the flucmation theorem in the context of thermostated systems [52-54], stochastic [55-59] as well as Hamiltonian [60] versions have been derived. A flucmation theorem has also been derived for nonequilibrium chemical reactions [62]. A closely related result is the nonequilibrium work theorem [61] which can also be derived from the microscopic Hamiltonian dynamics. 

An essential property of A is the existence of the inverse transformation A . This allows us to go back and forth between Hamiltonian dynamics and Markovian dynamics. In other words, A maps deterministic reversible dynamics to irreversible stochastic dynamics. 

The nonequilibrium equality in Eq. (16) becomes the nonequilibrium work relation originally derived by Jarzynski using Hamiltonian dynamics [31],... 

Kuchar, K. (1976) Dynamics of tensor fields in hyperspace, III. Journal of Mathematical Physics. 17(5) 801—20. (Discusses hypersurface dynamics of simple tensor fields with derivative gravitational coupling. The spacetime field action is studied and transformed into a hypersurface action. The hypersurface action of a covector field is cast into Hamiltonian form. Generalized Hamiltonian dynamics of spacetime hypertensors are discussed closing relations for the constraint functions are derived.)... 

Understanding chemical reactions has been a major preoccupation since the historical origins of chemistry. A main difficulty is to reconcile the macroscopic description in which reactions are rate processes ruling the time evolution of populations of chemical species with the microscopic Hamiltonian dynamics governing the motion of the translational, vibrational, and rotational degrees of freedom of the reacting molecules. 

The above scenario is accounted for by the normal form (4.9) truncated at fourth order in q with k = v = a = p = 0 and x < 0, taking p as the bifurcation parameter, which increases with energy (p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8) are given by p = 0 and dv/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. 

It can be seen that the solution of the problem of the energy-optimal guiding of the system from a chaotic attractor to another coexisting attractor requires the solution of the boundary-value problem (33)-(34) for the Hamiltonian dynamics. The difficulty in solving these problems stems from the complexity of the system dynamics near a CA and is related, in particular, to the delicate problems of the uniqueness of the solution, its behaviour near a CA, and the boundary conditions at a CA. 

Hamiltonian dynamics show that classical mechanics is invariant to ( t) and (t). In a macroscopic description of dissipative systems, we use collective variables of temperature, pressure, concentration, and convection velocity to define an instantaneous state. The evolution equations of the collective variables are not invariant under time reversal... 

Equation (21) makes it clear that the action-angle variable representation directly addresses the oscillation frequencies without looking into the details of the dynamics. Indeed, as shown below, the action-angle variable representation plays a key role in understanding important qualitative features of Hamiltonian dynamics. 

Hamiltonian dynamical system theory is the mathematical framework on which TST rests many textbooks, of various mathematical sophistication, describe this branch of pure/applied mathematics. Some of the various flavors are [20-24]. Very little of this vast information will be needed here, and we shall try to be as self-consistent as possible. 

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

Extension toward the fully nonlinear case is straightforward for 1-DOF Hamiltonians. The energy conservation relation H p,q) = E allows us to dehne (explicitly or implicitly) p = p q E), thereby reducing the ODE to a simple quadrature. In this procedure there is no problem of principle (unlike the n >2-DOE case). It works in practice also, and it is possible to adapt Eigs. 3-5 to the nonlinear regime. It must be underlined that besides that simple procedure, we present a theorem in dynamical system theory (containing Hamiltonian dynamics as a particular case). This theorem is valid for n DOEs (hence for n = 1) it relates the full dynamics to the linearized dynamics, called tangent dynamics in the mathematical literature. 

As a last point, it must be underlined that if the approximation of Hamiltonian dynamics is lifted— that is, if we include dissipation in one way or another—very little is known outside of either strong dissipation or 1-DOF systems. While this is outside of the scope of this review, the interested reader should consult Refs. 37 and 38. 

Description. Among all physically relevant Hamiltonian dynamics, one case is particularly important ... 

An introduction to Hamiltonian dynamics may be found in H. Goldstein, Classical Mechanics, Addison-Wesley, Reding, MA, 1964. 

K. R. Meyer and Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, Berlin, 1992. 

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