Langevin equation in systems with also negative temperatures

Baldovin, Marco; Puglisi, Andrea; Vulpiani, Angelo

doi:10.1088/1742-5468/aab687

Cited by 21 publications

(35 citation statements)

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“…[10], where cold atoms in an optical lattice display both positive and negative temperatures. It has also been studied theoretically, for instance see [11,21]. For the microscopic dynamics, the spins are started from a completely random configuration.…”

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Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures

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We consider the problem of building a continuous stochastic model, i.e. a Langevin or Fokker-Planck equation, through a well-controlled coarse-graining procedure. Such a method usually involves the elimination of the fast degrees of freedom of the "bath" to which the particle is coupled. Specifically, we look into the general case where the bath may be at negative temperatures, as foundfor instance-in models and experiments with bounded effective kinetic energy. Here, we generalise previous studies by considering the case in which the coarse-graining leads to (i) a renormalisation of the potential felt by the particle, and (ii) spatially dependent viscosity and diffusivity. In addition, a particular relevant example is provided, where the bath is a spin system and a sort of phase transition takes place when going from positive to negative temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive the Fokker-Planck equation from the microscopic dynamics. Our theoretical predictions show an excellent agreement with numerical simulations.

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Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures

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“…where ξ(t) is a white noise with unitary variance and D is a positive constant [31]. Since in the limit D = 0 the dynamics reduces to that of an isolated Hamiltonian system, we would like to find an algorithm that is symplectic in this limit.…”

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Statistical mechanics of systems with long-range interactions and negative absolute temperature

2019

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A Hamiltonian model living in a bounded phase space and with long-range interactions is studied. It is shown, by analytical computations, that there exists an energy interval in which the microcanonical entropy is a decreasing convex function of the total energy, meaning that ensemble equivalence is violated in a negative-temperature regime. The equilibrium properties of the model are then investigated by molecular dynamics simulations: first, the caloric curve is reconstructed for the microcanonical ensemble and compared to the analytical prediction, and a generalized Maxwell-Boltzmann distribution for the momenta is observed; then, the nonequivalence between the microcanonical and canonical descriptions is explicitly shown. Moreover, the validity of Fluctuation-Dissipation Theorem is verified through a numerical study, also at negative temperature and in the region where the two ensembles are nonequivalent.

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“…Approximate derivations for the case of non-linear forces [ 85 ] and for cases near non-equilibrium stationary processes [ 86 ] have also been recently considered. A particularly interesting test-case is that of systems where the Hamiltonian has a non-standard kinetic term, i.e., non-quadratic in the momentum, leading to non-trivial properties such as negative absolute temperature [ 87 , 88 , 89 , 90 ], a possibility recently verified also in experiments [ 91 ].…”

Section: Langevin Equation: When a Multiscale Structure Helpsmentioning

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“…In the following we discuss a very different approach, that is a practical procedure to build a Langevin equation from a long time series of data from experimental or numerical results. Up to our knowledge a similar procedure has been applied, previously, to many model systems [ 92 ] but only recently some of us have applied it to Hamiltonian systems, including systems displaying negative temperatures [ 90 ]. The procedure discussed here does not include a preliminary check for the validity of a LE description, it blindly assumes such a validity: Of course one then has to wonder in a critical way if such an assumption is correct, or at least, consistent.…”

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The Role of Data in Model Building and Prediction: A Survey Through Examples

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The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play-and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components. arXiv:1810.10446v1 [cond-mat.stat-mech] 24 Oct 2018The ideal formal model would be one which would cover the entire universe, which would agree with it in complexity, and which would have a one to one correspondence with it. Any one capable of elaborating and comprehending such a model in its entirety, would find the model unnecessary, because he could then grasp the universe directly as a whole. He would possess the third category of knowledge described by Spinoza. This ideal theoretical model cannot probably be achieved. Partial models, imperfect as they may be, are the only means for understanding the universe. This statement does not imply an attitude of defeatism but the recognition that the main tool of science is the human mind and that the human mind is finite.It is pretty impossible to detail the different procedures which had been followed to build the many models used in science, also because it can be safely claimed that there are not systematic protocols for model building. Possibly a lucky non-trivial exception is given by the framework of classical physics where a fairly clear approach exists. Once the forces are understood, one can write down proper differential equations that may be difficult to solve, but that always allow us to obtain reliable and useful results, for instance by means of qualitative or numerical analysis. Another example is when a reduced model is derived from a more general theory, for instance this the case of the Lorenz's model [4], which is obtained from a rather crude simplification of fluid dynamics equations. Such a model, in spite of its (apparent) simplicity, played a key role by allowing us to realize that many irregular motions encountered in nature can be due to chaos. This is a clear example of how even a "partial model" can help for understanding natural phenomena.The above approach is decidedly not applicable t...

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Langevin equation in systems with also negative temperatures

Cited by 21 publications

References 40 publications

Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures

Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures

Statistical mechanics of systems with long-range interactions and negative absolute temperature

The Role of Data in Model Building and Prediction: A Survey Through Examples

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