On the classical statistical mechanics of non-Hamiltonian systems

Tuckerman, Mark E.; Mundy, Christopher J.; Martyna, Glenn

doi:10.1209/epl/i1999-00139-0

Cited by 145 publications

(162 citation statements)

References 11 publications

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“…For example, this occurs when magnetic systems are modelled in terms of classical spins [4][5][6][7]. Deterministic methods [8][9][10], based on non-Hamiltonian dynamics [11][12][13][14][15][16][17][18][19], can sample the canonical distribution provided that the motion in the phase space of the relevant degrees of freedom is ergodic [1,3]. However, classical spin systems are usually formulated in terms of non-canonical variables [20,21], without a kinetic energy expressed through momenta in phase space, so that Nosé dynamics cannot be applied directly.…”

Section: Introductionmentioning

confidence: 99%

Bulgac-Kusnezov-Nosé-Hoover thermostats

Sergi

Ezra

2010

Phys. Rev. E

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In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in phase space by means of non-Hamiltonian brackets. Two generalized versions of the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are globally controlled by means of a single additional Nosé variable, and another where each demon is coupled to an independent Nosé-Hoover thermostat. Numerically stable and efficient measure-preserving time-reversible algorithms are derived in a systematic way for each case. The chaotic properties of the different phase space flows are numerically illustrated through the paradigmatic example of the one-dimensional harmonic oscillator. It is found that, while the simple Bulgac-Kusnezov thermostat is apparently not ergodic, both of the Nosé-Hoover controlled dynamics sample the canonical distribution correctly. * Electronic address: sergi@ukzn.ac.za † Electronic address: gse1@cornell.edu

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Section: Introductionmentioning

confidence: 99%

Bulgac-Kusnezov-Nosé-Hoover thermostats

Sergi

Ezra

2010

Phys. Rev. E

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“…The first term is the difference of the two kinetic energies (= m˙ x 2 1 /2 − m˙ x 2 2 /2), and the second term is K. The new Lagrangian (20) is unique up to terms nonlinear in x − and its time derivatives, which don't contribute to physical forces (see (11)). Using (11), or (9) and (10), gives the equations of motion in the physical limit, mẍ i = −αẋ i |˙ x | n−1 .…”

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confidence: 99%

“…These might include: developing partition functions for non-conservative statistical systems (see also [11]), studying the phase space structure of nonlinear dissipative dynamical systems, and developing variational numerical integrators for systems with physical dissipation, among others. Also, the appearance of a metric in (14), the hint of "covariance" in (12) and (13), and the use of doubled variables suggest additional structure for the symplectic manifold [12].…”

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confidence: 99%

Classical Mechanics of Nonconservative Systems

2013

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Hamilton's principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton's principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data. This subtlety can have undesirable effects. I present a formulation of Hamilton's principle that is compatible with initial value problems. Remarkably, this leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic non-conservative systems, thereby filling a longstanding gap in classical mechanics. Thus dissipative effects, for example, can be studied with new tools that may have application in a variety of disciplines. The new formalism is demonstrated by two examples of non-conservative systems: an object moving in a fluid with viscous drag forces and a harmonic oscillator coupled to a dissipative environment.Hamilton's principle of stationary action [1] is a cornerstone of physics and is the primary, formulaic way to derive equations of motion for many systems of varying degrees of complexity -from the simple harmonic oscillator to supersymmetric gauge quantum field theories. Hamilton's principle relies on a Lagrangian or Hamiltonian formulation of a system, which account for conservative dynamics but cannot describe generic non-conservative interactions. For simple dissipation forces local in time and linear in the velocities, one may use Rayleigh's dissipation function [1]. However, this function is not sufficiently comprehensive to describe systems with more general dissipative features like history-dependence, nonlocality, and nonlinearity that can arise in open systems.The dynamical evolution and final configuration of non-conservative systems must be determined from initial conditions. However, it seems under-appreciated that while initial data may be used to solve equations of motion derived from Hamilton's principle, the latter is formulated with boundary conditions in time, not initial conditions. This observation may seem innocuous, and it usually is, except that this subtlety may manifest undesirable features. Remarkably, resolving this subtlety opens the door to proper Lagrangian and Hamiltonian formulations of generic non-conservative systems.An illustrative example. To demonstrate the shortcoming of Hamilton's principle, consider a harmonic oscillator with amplitude q(t), mass m, and frequency ω coupled with strength λ to another harmonic oscillator with amplitude Q(t), mass M , and frequency Ω. The action for this system isThe total system conserves energy and is Hamiltonian but q(t) itself is open to exchange energy with Q and should thus be non-conservative. For a large number of Q oscillators the open (sub)system dynamics for q ought to be dissipative. Let us account for the effect of the Q oscillator on q(t) by finding solutions only to the equations of motion for Q and inse...

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“…In general, there is a phase-space compressibility factor κ associated with the lack of conservation of the measure that is given by minus the divergence of the flow in phase space. It may be shown that [23,24] …”

Section: Equations Of Motion With Constraintsmentioning

confidence: 99%

Discontinuous molecular dynamics for semiflexible and rigid bodies

Peña

Zon

Schofield

et al. 2007

The Journal of Chemical Physics

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A general framework for performing event-driven simulations of systems with semi-flexible or rigid bodies interacting under impulsive torques and forces is outlined. Two different approaches are presented. In the first, the dynamics and interaction rules are derived from Lagrangian mechanics in the presence of constraints. This approach is most suitable when the body is composed of relatively few point masses or is semi-flexible. In the second method, the equations of rigid bodies are used to derive explicit analytical expressions for the free evolution of arbitrary rigid molecules and to construct a simple scheme for computing interaction rules. Efficient algorithms for the search for the times of interaction events are designed in this context, and the handling of missed interaction events is discussed.

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On the classical statistical mechanics of non-Hamiltonian systems

Cited by 145 publications

References 11 publications

Bulgac-Kusnezov-Nosé-Hoover thermostats

Bulgac-Kusnezov-Nosé-Hoover thermostats

Classical Mechanics of Nonconservative Systems

Discontinuous molecular dynamics for semiflexible and rigid bodies

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