Surprisingly, the natural looking random walk leading to Brownian motion occurs to be often biased in a very subtle way: emphasizing some possibilities by only approximating maximal uncertainty principle. A new philosophy of stochastic modelling has been recently introduced, in which we use the only maximizing entropy choice of transition probabilities instead. Local behaviour of both approaches is similar, but they usually lead to completely different global situations. In contrast to Brownian motion leading to nearly uniform stationary density, this recent approach turns out in agreement with having strong localization properties, thermodynamical predictions of quantum mechanics, like thermalizing to dynamical equilibrium state of probability density as the quantum ground state: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case, or of the standard Schrödinger operator while including potential and making infinitesimal limit.
Surprisingly, the natural looking random walk leading to Brownian motion occurs to be often biased in a very subtle way: emphasizing some possibilities by only approximating maximal uncertainty principle. A new philosophy of stochastic modelling has been recently introduced, in which we use the only maximizing entropy choice of transition probabilities instead. Local behaviour of both approaches is similar, but they usually lead to completely different global situations. In contrast to Brownian motion leading to nearly uniform stationary density, this recent approach turns out in agreement with having strong localization properties, thermodynamical predictions of quantum mechanics, like thermalizing to dynamical equilibrium state of probability density as the quantum ground state: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case, or of the standard Schrödinger operator while including potential and making infinitesimal limit.
There are mainly used two basic approaches for probabilistic modeling of motion: stochastic in which the object literally makes succeeding random decisions using arbi-trarily chosen by us probabilities or ergodic in which we usually assume some chaotic classical evolution and probabilities appear while averaging over infinite trajectories. Both approaches assume we know the exact way the system evolves. In contrast, in this paper we will focus on thermodynamical motion models: assuming maximal uncertainty. Specifically, in the space of possible choices of transition probabil-ities, we take the optimizing entropy or free energy one. Equivalent condition appears to be calculating transition probabilities as proportions between single steps in canonical ensemble of trajectories going through a given point. It makes that these probabilities depend on the whole space- the walker cannot directly use them. This model is ther-modynamical: only we use it to predict the most probable behavior. Standard diffusion models like Brownian motion can be seen as obtained by locally maximizing uncertainty. For regular space it agrees with fully maximizing entropy choice of transition probabili-
We describe particles in a potential by a special diffusion process, the maximal entropy random walk (MERW) on a lattice. Since MERW originates in a variational problem, it shares the linear algebra of Hilbert spaces with quantum mechanics. The Born rule appears from measurements between equilibrium states in the past and the same equilibrium states in the future. Introducing potentials by the observation that time, in a gravitational field running in different heights with a different speed, MERW respects the rule that all trajectories of the same duration are counted with equal probability. In this way, MERW allows us to derive the Schrödinger equation for a particle in a potential and the Darwin term of the nonrelativistic expansion of the Dirac equation. Finally, we discuss why quantum mechanics cannot be simply a result of MERW, but, due to the many analogies, MERW may pave the way for further understanding.
As written by statistician George Box "All models are wrong, but some are useful", standard diffusion derivation or Feynman path ensembles use nonphysical infinite velocity/kinetic energy nowhere differentiable trajectories - what seems wrong, might be only our approximation to simplify mathematics. This article introduces some basic tools to investigate this issue. To consider ensembles of more physical finite velocity trajectories, we can work in (x,v) phase space like in Langevin equation with velocity controlling spatial steps, here also controlled with spatial potential V(x). There are derived and compared 4 approaches to predict stationary probability distributions: using Boltzmann ensemble of steps/points in space (GRW - generic random walk) or in phase space (psGRW), and analogously Boltzmann ensemble of paths in space (MERW - maximal entropy random walk) and in phase space (psMERW), also generalized to L{é}vy flights. Path ensembles generally have much stronger Anderson-like localization, MERW has stationary distribution exactly as quantum ground state. Proposed novel MERW in phase space has some slight differences, which might be distinguished experimentally. For example for 1D infinite potential well: ρ=1 stationary distribution for step ensemble, ρ∼sin2 for path ensemble (as in QM), and ρ∼sin for proposed smooth path ensembles - more frequently approaching the barriers due to randomly gained velocity.
Despite early developments on the foundations of quantum mechanics concern the wave function, quantum statistics has been developed with the density matrix formalism, leading to very important results in explaining molecular observations. Only recently, several authors argued a new interpretation by focusing on the wave function representing the quantum state of an isolated system, showing how a single wave function can exhibit statistical properties and generate the same results expected in the standard quantum statistical framework. Starting from these results, investigation on the foundations of quantum statistical mechanics has gained recently a renewed interest. As a matter of fact, the possibility of studying single molecule properties as well as the need of a better understanding of the effect of quantum dynamics, in order to develop new nanoscaled materials suitable to quantum computing tasks, have opened new intriguing questions leading to quantum statistical approaches far from being well understood and accepted. In this framework the behaviour of a single realization of quantum systems has gained a central role in the description of molecular systems. Furthermore, in recent years, an increasing number of studies has been presented on quantum dynamics through the numerical solution of the Schrödinger equation for systems of interacting components. These studies demonstrates that Quantum Dynamics Simulations could be a practicable route. In order to study phenomena such as dissipation, relaxation and thermalization, the focus has to be moved from isolated molecules to modular systems made of mutual interacting components, with model Hamiltonians possessing a sufficiently low dimensional representation. An important issue concerns the rules to be employed for the choice of the initial quantum state for the simulation of isolated systems. As long as one considers molecular degrees of freedom interacting with a (model) environment, there are no reasons to select a particular quantum state for the overall system and, therefore, a random choice has to be performed amongst a well defined statistical ensemble of pure states. Furthermore, one would like to operate a choice assuring that the simulation of the system is in a well defined thermal state with given temperature. This necessarily calls for a statistical description like for classical systems. A useful parametrization of the wave function will be presented in order to highlight the most appropriate variable for the statistical analysis. In particular some of them, called phases, retain all the dynamical information whereas the others, called populations, are the constants of motion. The latter, in particular are very important because they describe the equilibrium properties strictly related to the thermodynamic description. Since the dynamics of wave function does not supply any information about populations, the definition of a probability distribution on these variables is required. Different probability distributions on populations have been proposed, only on the bases of reasonable assumptions and they validation has been performed only with a posteriori considerations. In particular Fresch and Moro have demonstrated how the agreement with thermodynamics can be employed to discriminate different probability distributions on pure states. This had led the uniform ensembles to be, up to now, the most self-consistent models for quantum pure states. However, in this thesis I will highlight a drawback of the uniform distribution ensemble that can be described as follow: if we bring into contact two systems, even through a perturbative interaction, we are not able to describe the equilibrium properties after the interaction within the uniform distribution statistics, since the uniform character is lost. It represents a severe shortcoming of the statistical ensemble from a methodological point of view, since closed systems can be always considered as the result of interaction among previously isolated systems. On the other hand this drawback introduces a further requirement of a different nature that can be used for the definition of a new statistical ensemble. In this work I intend to find and characterize a statistical ensemble for populations that overcomes the drawbacks of the uniform distribution of pure states. The invariance of the thermal state in the coupling of identical systems will be used as a guideline in the definition af a new probability distribution on populations. Such an ensemble for pure states, called Thermalization Resilient Ensemble, provides a convenient framework for treating the interactions between quantum systems, as long as the structure of the statistical distribution is preserved and the identification of thermodynamic properties is assured. In perspective it should be the privileged statistical ensemble to implement Quantum Dynamics Simulations. Once the average properties of the Thermalization Resilient Ensemble have been introduced, I will obtain a probability distribution on pure states with the use of a geometrical analysis on the Hilbert space. The surface elements of an ellipsoidal manifold will be related to the probability density on populations. As a matter of fact the explicit form of the probability distribution is a prerequisite in order to perform Quantum Dynamical Simulations. However the results obtained through the geometrical analysis cannot be easily extended to systems with unbounded energy spectrum and an alternative strategy has been developed. A scaling algorithm on the basis of the uniform statistical ensemble will be described and this allows a well defined sampling of a probability distribution with desired averages. In this framework I demonstrate the emergence of thermodynamic behavior in the limit of macroscopic systems. In the last part of the thesis I consider the dynamical features of the thermalization experiment. Two identical systems, initially at different temperature, will be brought in interaction and the analysis of the final equilibrium state will be performed for two different generic forms of the interaction Hamiltonian, highlighting how the statistical approach can be very useful in the definition of the equilibrium in complex quantum systems.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.
Extremization of entropy, usually subject to an average energy constraint,(Maxwell-Boltzmann gas) is often associated with maximization of the number of microstates N gas particles may represent. The math procedure of extremization (variational calculus) is then associated with the ln of a factorial expression which reduces to n(ei)ln(n(ei)) using Stirling’s approximation. The central limit theorem, on the other hand, states that the distribution of sample means of a random variable approaches a Gaussian, i.e exp(-pp/(2sig sig)). Here sig sig is the standard deviation. For a random variable, the important or characteristic information is the standard deviation associated with the average of the square pp. We suggest here that an alternative way of looking at maximization of entropy (which is a stability statement we argue) other than ln of permutations is to consider the stability of the variation of the variance of pp subject to a change in the probability, with pp written in terms of P(p). Average kinetic energy in a statistical problem represents such a variance and should be invariant under changes of the probability subject to an average kinetic energy constraint, we suggest. How does this idea apply to quantum mechanics? In quantum mechanics, one has a statistical function exp(ipx) and for bound states creates W(x)=wavefunction = Sum over p a(p)exp(ipx). a(p)a(p) is formally the probability for p, but this does not mean that a(p)a(p) is a random variable because one has the constraint: Sum over p a(p)a(p) pp2/m + Integral dx V(x) W(x)W(x) = En. In the case of a harmonic oscillator, however, one knows a priori that the two summands are each equal to En/2. Furthermore there are a set of En values, starting at a ground state and increasing. All En values, except the ground state, are metastable. Thus the ground state is a special state. Given that Sum over p a(p)a(p)pp2/m = Eo/2, a priori, one may consider that all of the information of a(p)a(p) is associated with Eo. This is exactly the case for a Gaussian from the central limit theorem. Alternatively, from information theory, information = ln(probability) which yields the same result. If this is the case, then pp/2m a(p)a(p) written strictly in terms of P(p)=a(p)a(p) should be stationary upon changing P(p) subject to the E/2 constraint. This special situation, however, only holds for the ground state oscillator. What about average potential energy for the ground state oscillator? If the probability for p (momentum) is a Gaussian due to the Central Limit theorem, this means that the standard deviation is the information (and is linked with randomness). Taking a Fourier transform of a(p) cannot add more information to the picture, we argue. W(x)W(x) should also be a Gaussian, as it is. As a result, xxW(x)W(x), written in terms of P(x)=W(x)W(x) subject to the same Eo/2 constraint should represent a maximization or stationary situation, we argue.