Resolution of Loschmidt’s paradox: The origin of irreversible behavior in reversible atomistic dynamics

Holian, Brad Lee; Hoover, William G.; Posch, Harald A.

doi:10.1103/physrevlett.59.10

Cited by 196 publications

(141 citation statements)

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“…Through the appearance of the Lyapunov exponents, it establishes a link between transport theory and dynamical systems theory and opens a new way to understand irreversibility and the Second Law [98]. It should be noted that this insight was first obtained by computer simulations [93] with theoretical verification for simple cases following later [99]. We like to think that Boltzmann would have approved of these results which extend his way of thinking to a wide class of dynamical systems in nonequilibrium stationary states.…”

mentioning

confidence: 94%

“…Nevertheless, a computer simulation of a thermostated nonequilibrium system always gives a positive time-averaged friction variable (or a positive sum of time-averaged friction variables, if more than one thermostat is involved), if the evolution is followed long enough either in the positive or negative direction of time. The system always behaves irreversibly on a macroscopic time scale in accordance with the Second Law although the microscopic dynamics is time reversible [93,94,95,96].To understand this behaviour, we consider as a simple example the Lorentz gas introduced in Sec. 3.…”

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confidence: 99%

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Realizing Boltzmann's dream: computer simulations in modern statistical mechanics

Dellago

Posch

2008

ESI Lectures in Mathematics and Physics

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Abstract. Today, computer simulations belong to the most important and powerful theoretical tools for the investigation of statistical mechanical systems particularly in the field of condensed matter. Many simulation methods may directly, or indirectly, be traced back to ideas of Ludwig Boltzmann or are used to carry on the program he had envisioned. Using several illustrative examples, we discuss the role of computer simulation in modern statistical mechanics and, in particular, its relation to Boltzmann's legacy.1 What if? What would Boltzmann have done with a computer? Of course, any answer to this question is highly speculative, but it is easy to imagine that Boltzmann 2 C. Dellago and H. A. Posch would have realized what a wonderful tool for scientific research and discovery the computer is, particularly in his field of statistical mechanics. Most likely, Boltzmann would have invented the molecular dynamics method and used it to test and further develop the molecular models of matter he and his contemporaries created. All he required he already knew, and his motivation is more than clear from his work. With such simulations, Boltzmann literally could have watched how a system relaxes towards equilibrium, and he could have performed a numerical analysis of the validity of the assumption of molecular chaos, which is so central to his kinetic equation. Certainly, he would have determined the equations of state of dilute gases and of dense liquids and solids. One may further speculate that Boltzmann would have made creative use of modern visualization tools that, today, are of such crucial importance in simulation studies and provide insight and guidance not availiable otherwise. It is also amusing to think about what Boltzmann would not have done, if he had had access to a computer. For instance, would Boltzmann have bothered to write down the Boltzmann equation? Perhaps he would just have run a molecular dynamics simulation for hard spheres with simple collision rules to follow the dynamics of his model gas. From such a simulation he could have calculated properties of dilute and dense gases in order to compare them with experimental data. Then, the need to write down an approximate and complicated integro-differential equation that cannot even be solved analytically except for very simple cases would not have arisen. Or would Boltzmann have tried to develop a virial expansion for the hard sphere gas if he could have determined the equation of state with high precision from simulations? Nobody knows, but statistical mechanics might have unfolded in a completely different way, if computers had been available at Boltzmann's time. While it is not hard to imagine where Boltzmann would have begun his computational investigation, it is impossible to predict where insights gleaned from simulations would have taken a mind like his.In this article we will take a more modest attitude and reflect on the significance of computer simulations in the research program initiated by Boltzmann and his contemporaries. Since t...

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confidence: 94%

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confidence: 99%

Realizing Boltzmann's dream: computer simulations in modern statistical mechanics

Dellago

Posch

2008

ESI Lectures in Mathematics and Physics

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“…Though the microscopic equations of motion of these systems are time-reversible the macroscopic dynamics is irreversible in nonequilibrium leading to momentum and energy fluxes with well-defined transport coefficients [5,6,[10][11][12][13][14], which appears to be a paradox. However, investigations of the microscopic dynamics with methods from dynamical system theory could resolve this paradox by showing that the microscopic dynamics is nonlinear and highly unstable [12,14] and leads to a phase space volume contraction onto a fractal attractor [11,15,16]. From the analysis of conventional thermostats, further relations between quantities characterizing the microscopic dynamics and quantities characterizing macroscopic transport could be established.…”

Section: Introductionmentioning

confidence: 99%

“…Conventional types of them, such as the Gaussian and the Nosé-Hoover thermostat, are based on introducing a momentum dependent friction coefficient into the microscopic equations of motion [5][6][7][8][9]. Though the microscopic equations of motion of these systems are time-reversible the macroscopic dynamics is irreversible in nonequilibrium leading to momentum and energy fluxes with well-defined transport coefficients [5,6,[10][11][12][13][14], which appears to be a paradox. However, investigations of the microscopic dynamics with methods from dynamical system theory could resolve this paradox by showing that the microscopic dynamics is nonlinear and highly unstable [12,14] and leads to a phase space volume contraction onto a fractal attractor [11,15,16].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering

Rateitschak

Klages

2002

Phys. Rev. E

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In recent work a deterministic and time-reversible boundary thermostat called thermostating by deterministic scattering has been introduced for the periodic Lorentz gas [Phys. Rev. Lett. 84, 4268 (2000)]. Here we assess the nonlinear properties of this new dynamical system by numerically calculating its Lyapunov exponents. Based on a revised method for computing Lyapunov exponents, which employs periodic orthonormalization with a constraint, we present results for the Lyapunov exponents and related quantities in equilibrium and nonequilibrium. Finally, we check whether we obtain the same relations between quantities characterizing the microscopic chaotic dynamics and quantities characterizing macroscopic transport as obtained for conventional deterministic and time-reversible bulk thermostats.

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“…The entropy itself is undefined in a nonequilibrium process * pkpatra@vt.edu † rbatra@vt.edu because of the multifractal nature of the phase space and the consequent divergence of log(f ), f being the probability distribution function [7,12,13]. Apart from the thermodynamic definition, temperature may be expressed in several different ways, with each definition having its own sets of problems in nonequilibrium settings.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium temperature measurement in a thermal conduction process

Patra

Batra

2017

Phys. Rev. E

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We identify the temperature being measured by a thermometer in a nonequilibrium scenario by studying heat conduction in a three-dimensional Lennard-Jones (LJ) system whose two ends are kept at different temperatures. It is accomplished by modeling the thermometer particles also with the LJ potential but with added tethers to prevent their rigid body motion. These models of the system and the thermometer mimic a real scenario in which a mechanical thermometer is "inserted" into a system and kept there long enough for the temperature to reach a steady value. The system is divided into five strips, and for each strip the temperature is measured using an embedded thermometer. Unlike previous works, these thermometers are small enough not to alter the steady state of the nonequilibrium system. After showing initial transients, the thermometers eventually show steady-state conditions with the subregions of the system and provide values of the different temperature definitions-kinetic, configurational, dynamical, and higher-order configurational. It is found that their kinetic and the configurational temperatures are close to the system's kinetic temperature except in the two thermostatted regions. In the thermostatted regions, where the system's kinetic and the configurational temperatures are significantly different, the thermometers register a temperature substantially different from either of these two values. With a decrease in the system density and size, these differences between the kinetic and the configurational temperatures of the thermometer become more pronounced.

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Resolution of Loschmidt’s paradox: The origin of irreversible behavior in reversible atomistic dynamics

Cited by 196 publications

References 10 publications

Realizing Boltzmann's dream: computer simulations in modern statistical mechanics

Realizing Boltzmann's dream: computer simulations in modern statistical mechanics

Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering

Nonequilibrium temperature measurement in a thermal conduction process

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