Session X40: General Statistical and Nonlinear Physics

Chaotic dynamics of a candle oscillator

The candle oscillator is a simple, fun experiment dating to the late nineteenth century. It consists of a candle with a rod that is transverse to its long axis, around which it is allowed to pivot. When both ends of the candle are lit, an oscillatory motion will initiate due to different mass loss as a function of the flame angle.  Stable oscillations can develop due to damping when the system has friction between the rod and the base where the rod rests. However, when friction is minimized, it is possible for chaos to develop.  In this talk we will show periodic orbits found in the system as well as calculated, maximal Lyapunov exponents. We show that the system can be described by three ordinary differential equations (one each for angle, angular velocity and mass loss) that can reproduce the experimental data and the transition from stable oscillations to chaotic dynamics as a function of damping.   

Spike Bursts from an Excitable Optical System

Diode Lasers with double optical feedback are shown to present power drop spikes with statistical distribution controllable by the ratio of the two feedback times. The average time between spikes and the variance within long time series are studied. The system is shown to be excitable and present bursting of spikes created with specific feedback time ratios and strength. A rate equation model, extending the Lang-Kobayashi single feedback for semiconductor lasers proves to match the experimental observations. Potential applications to construct network to mimic neural systems having controlled bursting properties in each unit will be discussed.   

Singular probability distribution of a parametric oscillator driven by Poisson noise

We provide the results of the theoretical and experimental studies of the probability distribution of a parametric oscillator, which is additionally driven by a Poisson-like noise. The noise consists of pulses at the vibration frequency with duration small compared to the oscillator relaxation time but long compared to the vibration period. We find that the stationary probability distribution of an oscillator quadrature can display a self-similar structure of sharp peaks, almost symmetrical with respect to the maximum, or can have a strongly asymmetric two-peak structure. The form of the distribution depends on the oscillator dynamics in the rotating frame and the rate of the noise pulses. In particular, the self-similar multi-peak structure emerges if the oscillator dynamics in the rotating frame is underdamped. The peaks have a singular power-law shape. We show that the singularity is smeared by thermal noise, which makes the peaks Gaussian near the maxima. We also discuss the frequently encountered situation where the Poisson noise describes fluctuations of the oscillator eigenfrequency. The theoretical and experimental results are in excellent agreement.   

Quantum Boltzmann Machine

The field of machine learning has been revolutionized by the recent improvements in the training of deep networks. Their architecture is based on a set of stacked layers of simpler modules. One of the most successful building blocks, known as a restricted Boltzmann machine, is an energetic model based on the classical Ising Hamiltonian. In our work, we investigate the benefits of quantum effects on the learning capacity of Boltzmann machines by extending its underlying Hamiltonian with a transverse field. For this purpose, we employ exact and stochastic training procedures on data sets with physical origins.   

Quantum Feynman Ratchet

As nanotechnology advances, understanding of the thermodynamic properties of small systems becomes increasingly important. Such systems are found throughout physics, biology, and chemistry manifesting striking properties that are a direct result of their small dimensions where fluctuations become predominant. The standard theory of thermodynamics for macroscopic systems is powerless for such ever fluctuating systems. Furthermore, as small systems are inherently quantum mechanical, influence of quantum effects such as discreteness and quantum entanglement on their thermodynamic properties is of great interest. In particular, the quantum fluctuations due to quantum uncertainty principles may play a significant role. In this talk, we investigate thermodynamic properties of an autonomous quantum heat engine, resembling a quantum version of the Feynman Ratchet, in non-equilibrium condition based on the theory of open quantum systems. The heat engine consists of multiple subsystems individually contacted to different thermal environments.   

Reversibility in Quantum Models of Stochastic Processes

Natural phenomena such as time series of neural firing, orientation of layers in crystal stacking and successive measurements in spin-systems are inherently probabilistic. The provably minimal classical models of such stochastic processes are $\varepsilon $-machines, which consist of internal states, transition probabilities between states and output values. The topological properties of the $\varepsilon $-machine for a given process characterize the structure, memory and patterns of that process. However $\varepsilon $-machines are often not ideal because their statistical complexity (C$_{\mathrm{\mu }})$ is demonstrably greater than the excess entropy (\textbf{E}) of the processes they represent. Quantum models (q-machines) of the same processes can do better in that their statistical complexity (C$_{\mathrm{q}})$ obeys the relation C$_{\mathrm{\mu }}\ge $C$_{\mathrm{q}}\ge $\textbf{E}. q-machines can be constructed to consider longer lengths of strings, resulting in greater compression. With code-words of sufficiently long length, the statistical complexity becomes time-symmetric -- a feature apparently novel to this quantum representation. This result has ramifications for compression of classical information in quantum computing and quantum communication technology.   

Finding stability domains and escape rates in kicked Hamiltonians

We use an effective Hamiltonian to characterize particle dynamics and find escape rates in a one dimensional system with a periodically kicked Hamiltonian. We study a model of particles in storage rings which is given by a symplectic map where the chaos is described by the KAM theorem. Ignoring the resonances, the dynamics typically has a finite region in phase space where it is stable. Photon noise in the system leads to particle loss from this stable region. Determining this `aperture' and finding escape rates is therefore an important physical problem. We characterize the stable region in phase space using a perturbation theory developed in the context of quantum mechanics. We then derive analytical expressions for the escape rate in the small damping regime and compare them with numerical simulations. We discuss the possibility of extending the procedure to include higher dimensions and more complicated noise terms.   

The canonical ensemble revisited: a projection operator approach

Constraining the particle number N in the canonical ensemble hampers the systematic calculation of the partition function Z$_N$ for non-interacting fermions and bosons, unlike in the case of the grand-canonical ensemble. Recently, we have shown that this task can be accomplished by invoking a projection operator that automatically imposes the particle number constraint in the many-particle Hilbert space. As a result, an integral representation is obtained for Z$_N$, as well as for the the two-point and four-point correlation functions. As an illustration, the Helmholtz free energy and the chemical potential are calculated for a two-dimensional electron gas typically residing in the inversion layer of a field-effect transistor.   

Novel dynamics and thermodynamics of a new Hamiltonian mean field model

Statistical systems are idealized by the hypothesis that the particles do not interact among them, or the range of interactions is short enough, reaching very fast the statistical state that we know as equilibrium. However, systems with long-range interactions are common in nature because of they are observed from the atomic scale to the astronomical scale, exhibiting some anomalies as inequivalence of ensembles, negative heat capacity, ergodicity breaking, non equilibrium phase transitions, quasi-stationarity, anomalous diffusion, etc. We present in this contribution a new Hamiltonian mean field model whose potential is inspired in the dipole-dipole interactions. The equilibrium is analytically studied in the canonical ensemble and coincides with the one obtained from molecular dynamics simulations (microcanonical ensemble). We notice, this model presents a kind of inequivalence of ensembles in long-standing states before arriving at equilibrium. However, the novelty, compared to other models presented in recent literature, is that two quasi-stationary states appear in the behavior of this system. The first quasi-stationary state decays to a second one, which is different to the first, before going to the equilibrium.We characterize them by its dynamics and thermodynamics.   

Optimization of finite-size errors in finite-temperature calculations of unordered phases

It is common knowledge that the microcanonical, canonical, and grand canonical ensembles are equivalent in thermodynamically large systems. Here, we study finite-size effects in the latter two ensembles. We show that contrary to naive expectations, finite-size errors are exponentially small in grand canonical ensemble calculations of translationally invariant systems in unordered phases at finite temperature. Open boundary conditions and canonical ensemble calculations suffer from finite-size errors that are only polynomially small in the system size. We further show that finite-size effects are generally smallest in numerical linked cluster expansions. Our conclusions are supported by analytical and numerical analyses of classical and quantum systems.   

Complex Pole Approach in Thermodynamic Description of Fluid Mixtures with Small Number of Molecules

Physically consistent description of equilibrium small molecular systems requires the extension of thermodynamics. The reason is the absence of thermodynamic limit, which is mandatory for the applicability of classical thermodynamics. New theoretical method of complex pole decomposition for the statistical description of small multicomponent molecular systems is implemented. Similar approach has been previously developed and applied in nuclear physics for finite systems of nucleons. We have significantly transformed and extended the original formulation to make it work for multicomponent molecular mixtures in small systems. The aim of this research is to provide new comprehensive description of small equilibrium molecular systems with numerous scientific and industrial applications for artificial and natural materials with nanopores. Several cases for molecular systems in small cavities are studied. In particular size-dependent additional pressure for small systems is evaluated analytically and numerically. The obtained results are in correspondence to published experimental data and molecular dynamics simulations.   

Exact Phi4 Critical Exponents via the Limit of Finite Periodic Systems

We formulate an RG procedure to nonperturbatively calculate the critical exponents of phi4$^{\mathrm{\thinspace }}$theory in arbitrary dimension. Our method first calculates the exact RG equations for a finite but arbitrarily large system with periodic boundary. We then take the limit as that boundary diverges to simplify the equations and recover a true critical point of the system. In particular this provides the 3d critical Ising exponents to high precision. This method is not specific to phi4$^{\mathrm{\thinspace }}$theory and thus should apply to many other systems.   

The three-dimensional $O(n\to\infty)$ $\phi^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover

The $O(2)$ $\phi^4$ model on a 3D film of thickness $L$ with free boundaries is relevant for the explanation of the thinning of wetting layers of ${}^4$He caused by critical Casimir forces near and below the $\lambda$-transition. Just as its $O(n)$ analog, the model has long-range order below the bulk critical temperature $T_c$ if $L=\infty$, but remains disordered for all $T>0$ when $L<\infty$. A proper analysis of its scaling behavior near $T_c$ is challenging: it involves a nontrivial dimensional crossover in addition to bulk, boundary, and finite-size critical behaviors. The $n\to\infty$ limit of the model can be solved exactly in terms of the eigenvalues and eigenfunctions of a self-consistent Schr\"odinger equation whose potential $v(z)$ becomes singular at the boundary planes. Complementing recent numerically exact results, we derive various exact analytical results for series expansion coefficients of $v(z)$, its $L=\infty$ scattering data for all values $m\gtreqless 0$ of the termperature scaling field, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force using a combination of boundary-operator and short-distance expansions, proper extensions of inverse scattering theory, new trace formulae, and semi-classical expansions.   

A Molecular Model for Chiral Symmetry Breaking

In this work, we present a new class of molecular models for chiral phenomena in condensed matter systems. A key feature of these models is the ability of the four-site (tetramer) ``molecules'' to inter-convert between two distinct chiral forms (enantiomers). Given this feature, we use analytical theory and computer simulations to investigate the emergent chiral properties (including symmetry breaking) over a range of conditions. In particular, we consider the single-molecule level and condensed-phase behavior of our model system. Interestingly, we find that our liquid-phase predictions are in excellent agreement with recent experimental reports on chiral self-sorting in isotropic liquids. From this perspective, our model demonstrates accurate predictive capabilities, as well as a platform for understanding the microscopic origins of a variety of chiral phenomena. In a broader context, we anticipate that this class of models will be relevant to chirality-dominated areas such as the pharmaceutical industry and pre-biotic geochemistry.   

Thermal diffusion and colored energy dissipation in hydrogen bonded liquids.

H-bonded liquids show a manifold energy dissipation dynamics due to: strong directionality of H-bonds and complexity of their network. This affects both thermal diffusion and energy dissipation mechanisms in pump-probe spectroscopy experiments. By nonequilibrium molecular dynamics (MD) simulations we investigate such phenomena in liquid methanol. While heat transport is studied by approach-to-equilibrium MD, energy dissipation is investigated by making use of a novel Generalized Langevin Equation (GLE) colored noise thermostat, which can generate a non-equilibrium frequency-resolved dynamics by using a correlated noise. The colored thermostat can thermally excite a narrow range of vibrational modes, typically the stretching mode of the OH involved in H-bonding, leaving the other degrees of freedom at the equilibrium temperature. The energy dissipation is then observed as a function of time, by probing the excitation decay and the energy transfer to other modes. In particular, by monitoring in time the different contributions to the potential energy of the system, we evaluate how energy is transferred from the excited mode to other modes of the nearby molecules and provide understanding on the dynamics of H-bonded liquids, as resulting from current experimental investigations