56 research outputs found
Statistical mechanics of systems with long-range interactions and negative absolute temperature
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
Physical interpretation of the canonical ensemble for long-range interacting systems in the absence of ensemble equivalence
In systems with long-range interactions, since energy is a non-additive
quantity, ensemble inequivalence can arise: it is possible that different
statistical ensembles lead to different equilibrium descriptions, even in the
thermodynamic limit. The microcanonical ensemble should be considered the
physically correct equilibrium distribution as long as the system is isolated.
The canonical ensemble, on the other hand, can always be defined
mathematically, but it is quite natural to wonder to which physical situations
it does correspond. We show numerically and, in some cases, analytically, that
the equilibrium properties of a generalized Hamiltonian mean-field model in
which ensemble inequivalence is present are correctly described by the
canonical distribution in (at least) two different scenarios: a) when the
system is coupled via local interactions to a large reservoir (even if the
reservoir shows, in turn, ensemble inequivalence) and b) when the mean-field
interaction between a small part of a system and the rest of it is weakened by
some kind of screening
Langevin equations from experimental data: the case of rotational diffusion in granular media
A model has two main aims: predicting the behavior of a physical system and
understanding its nature, that is how it works, at some desired level of
abstraction. A promising recent approach to model building consists in deriving
a Langevin-type stochastic equation from a time series of empirical data. Even
if the protocol is based upon the introduction of drift and diffusion terms in
stochastic differential equations, its implementation involves subtle
conceptual problems and, most importantly, requires some prior theoretical
knowledge about the system. Here we apply this approach to the data obtained in
a rotational granular diffusion experiment, showing the power of this method
and the theoretical issues behind its limits. A crucial point emerged in the
dense liquid regime, where the data reveal a complex multiscale scenario with
at least one fast and one slow variable. Identifying the latter is a major
problem within the Langevin derivation procedure and led us to introduce
innovative ideas for its solution
Irreversibility and typicality: A simple analytical result for the Ehrenfest model
With the aid of simple analytical computations for the Ehrenfest model, we
clarify some basic features of macroscopic irreversibility. The stochastic
character of the model allows us to give a non-ambiguous interpretation of the
general idea that irreversibility is a typical property: for the vast majority
of the realizations of the stochastic process, a single trajectory of a
macroscopic observable behaves irreversibly, remaining "very close" to the
deterministic evolution of its ensemble average, which can be computed using
probability theory. The validity of the above scenario is checked through
simple numerical simulations and a rigorous proof of the typicality is provided
in the thermodynamic limit
Mesoscopic virial equation for nonequilibrium statistical mechanics
We derive a class of mesoscopic virial equations governing energy partition
between conjugate position and momentum variables of individual degrees of
freedom. They are shown to apply to a wide range of nonequilibrium steady
states with stochastic (Langevin) and deterministic (Nos\'e--Hoover) dynamics,
and to extend to collective modes for models of heat-conducting lattices. A
generalised macroscopic virial theorem ensues upon summation over all degrees
of freedom. This theorem allows for the derivation of nonequilibrium state
equations that involve dissipative heat flows on the same footing with state
variables, as exemplified for inertial Brownian motion with solid friction and
overdamped active Brownian particles subject to inhomogeneous pressure.Comment: 14 pages, 3 figures. Some revision
Scaling symmetry, renormalization, and time series modeling
We present and discuss a stochastic model of financial assets dynamics based
on the idea of an inverse renormalization group strategy. With this strategy we
construct the multivariate distributions of elementary returns based on the
scaling with time of the probability density of their aggregates. In its
simplest version the model is the product of an endogenous auto-regressive
component and a random rescaling factor designed to embody also exogenous
influences. Mathematical properties like increments' stationarity and
ergodicity can be proven. Thanks to the relatively low number of parameters,
model calibration can be conveniently based on a method of moments, as
exemplified in the case of historical data of the S&P500 index. The calibrated
model accounts very well for many stylized facts, like volatility clustering,
power law decay of the volatility autocorrelation function, and multiscaling
with time of the aggregated return distribution. In agreement with empirical
evidence in finance, the dynamics is not invariant under time reversal and,
with suitable generalizations, skewness of the return distribution and leverage
effects can be included. The analytical tractability of the model opens
interesting perspectives for applications, for instance in terms of obtaining
closed formulas for derivative pricing. Further important features are: The
possibility of making contact, in certain limits, with auto-regressive models
widely used in finance; The possibility of partially resolving the long-memory
and short-memory components of the volatility, with consistent results when
applied to historical series.Comment: Main text (17 pages, 13 figures) plus Supplementary Material (16
pages, 5 figures
Option pricing with non-Gaussian scaling and infinite-state switching volatility
Volatility clustering, long-range dependence, and non-Gaussian scaling are
stylized facts of financial assets dynamics. They are ignored in the Black &
Scholes framework, but have a relevant impact on the pricing of options written
on financial assets. Using a recent model for market dynamics which adequately
captures the above stylized facts, we derive closed form equations for option
pricing, obtaining the Black & Scholes as a special case. By applying our
pricing equations to a major equity index option dataset, we show that
inclusion of stylized features in financial modeling moves derivative prices
about 30% closer to the market values without the need of calibrating models
parameters on available derivative prices.Comment: Revised version. 31 pages, 4 figure
Derivation of a Langevin equation in a system with multiple scales: the case of negative temperatures
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 found, for instance, in models and experiments with bounded
effective kinetic energy. Here, we generalize previous studies by considering the case in which the coarse
graining leads to (i) a renormalization 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 excellent agreement with numerical simulation
The Role of Data in Model Building and Prediction: A Survey Through Examples
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
Using machine-learning modelling to understand macroscopic dynamics in a system of coupled maps
Machine learning techniques not only offer efficient tools for modelling
dynamical systems from data, but can also be employed as frontline
investigative instruments for the underlying physics. Nontrivial information
about the original dynamics, which would otherwise require sophisticated ad-hoc
techniques, can be obtained by a careful usage of such methods. To illustrate
this point, we consider as a case study the macroscopic motion emerging from a
system of globally coupled maps. We build a coarse-grained Markov process for
the macroscopic dynamics both with a machine learning approach and with a
direct numerical computation of the transition probability of the
coarse-grained process, and we compare the outcomes of the two analyses. Our
purpose is twofold: on the one hand, we want to test the ability of the
stochastic machine learning approach to describe nontrivial evolution laws, as
the one considered in our study; on the other hand, we aim at gaining some
insight into the physics of the macroscopic dynamics by modulating the
information available to the network, we are able to infer important
information about the effective dimension of the attractor, the persistence of
memory effects and the multi-scale structure of the dynamics.Comment: 17 pages, 13 figure
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