Quantum and classical statistical mechanics

The current and voltage fluctuations in a normal tunnel junction are calculated from microscopic theory. The power spectrum can deviate from the familiar Johnson-Nyquist form when the self-capacitance of the junction is small, at low temperatures (C≲0.1 pF, T≲10 mK), ...

We present experimental and theoretical results on highly excited Rydberg atoms passing through a waveguide. The waveguide is excited in a coherent mode with a superimposed component of technically generated noise. In the theoretical part of the paper we derive and solve a master equation for a Rydberg atom driven by a monochromatic coherent microwave field in the presence of noise. We show that a Rydberg atom subjected to a mixture of coherent modes and noise fields exhibits four dynamical regimes: (i) diffusive broadening, (ii) localization, (iii) destruction of coherence and localization, and (iv) relaxation to equilibrium. The four regimes are passed one after the other as a function of irradiation time. They occur on different time scales and are thus temporally well separated from each other. The theory is checked by an experiment on the time dependence of the population distribution of highly excited rubidium Rydberg atoms initially prepared in a unique and well-defined Rydberg state and irradiated by a strong microwave field. The localization regime, characterized by a "freezing" of the width of the wave packet with respect to the Rydberg levels, has been observed. The addition of a small noise component was shown to lead to delocalization after times inversely proportional to the noise power, as predicted by our theory.

In this work, our prime focus is to study the one to one correspondence between the conduction phenomena in electrical wires with impurity and the scattering events responsible for particle production during stochastic inflation and reheating implemented under a closed quantum mechanical system in early universe cosmology. In this connection, we also present a derivation of quantum corrected version of the Fokker-Planck equation without dissipation and its fourth order corrected analytical solution for the probability distribution profile responsible for studying the dynamical features of the particle creation events in the stochastic inflation and reheating stage of the universe. It is explicitly shown from our computation that quantum corrected Fokker-Planck equation describe the particle creation phenomena better for Dirac delta type of scatterer. In this connection, we additionally discuss Itô, Stratonovich prescription and the explicit role of finite temperature effective potential for solving the probability distribution profile. Furthermore, we extend our discussion of particle production phenomena to describe the quantum description of randomness involved in the dynamics. We also present computation to derive the expression for the measure of the stochastic non-linearity (randomness or chaos) arising in the stochastic inflation and reheating epoch of the universe, often described by Lyapunov Exponent. Apart from that, we quantify the quantum chaos arising in a closed system by a more strong measure, commonly known as Spectral Form Factor using the principles of random matrix theory (RMT). Additionally, we discuss the role of out of time order correlation function (OTOC) to describe quantum chaos in the present non-equilibrium field theoretic setup and its cona e-mails:

We adopt a formulation of the Mach principle that the rest mass of a particle is a measure of it's long-range collective interactions with all other particles inside the horizon. As a consequence, all particles in the universe form a 'gravitationally entangled' statistical ensemble and one can apply the approach of classical statistical mechanics to it. It is shown that both the Schrödinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to the discreetness and finiteness of the Machian ensemble.

A superluminal quantum-vortex model of the electron and the positron is produced from a superluminal double-helix model of the photon during electron-positron pair production. The two oppositely-charged (with Q = ±e sqrt (2/α) = 16.6e) open-helix spin-½ half-photons compose the double-helix photon. These half-photons separate and curl up their separated superluminal single-helical trajectories to form an electrically-charged superluminal closed-helix spin-½ quantum-vortex electron model and a corresponding positron model. The helical radius and the Dirac equation's zitterbewegung angular frequency of the quantum vortex electron and positron models equal the helical radius and zitterbewegung angular frequency of the two spin-½ half-photons, each of energy E = mc^2 , that composed the double-helix photon model of energy E = 2mc^2 from which the electron and positron models were produced. The photon and electron models are also compatible when a photon of energy E > 2mc^2 produces a relativistic electron-positron pair. Implications of the quantum vortex electron model for electron stability are discussed.

We examine the equilibrium solutions of the BCS theory of superconductivity in the low temperature limit, allowing the attraction band to be asymmetric with respect to the chemical potential of the system µR. If we denote by µ the middle of the attraction band, we observe that the super-conducting phase is formed only if |µR − µ| < 2∆0, where ∆0 is the energy gap at zero temperature in the standard BCS theory. If |µR − µ| < 2∆0, the system of equations which give the energy gap has two solutions for each value of µR − µ: one with ∆(T = 0) = ∆0 and another one, with ∆(T = 0) < ∆0 (∆(T = 0) is the energy gap at 0 K). If 0 < |µR − µ| < 2∆0, a quasiparticle imbalance appears in equilibrium. In the " standard BCS limit, " which is µR → µ, beside the standard solution ∆(T = 0) = ∆0, we find another one, with ∆(T = 0) = ∆0/3 and nonzero quasiparticle population. The formalism takes into account in a consistent way the variation of the total number of particles with the population of the quasiparticle states.

"Networks are mathematically directed (in practical applications also undirected) graphs and a graph is a one-dimensional abstract complex, i.e., a topological space. Network theory focuses on various topological structures and properties, dynamic properties, and functionality-topology relationship, etc. There are some common mathematical foundations, theories and methodology for network analysis, in which graph theory, statistics, and operational research, etc., are the fundamental sciences of network analysis. Various emerging biological networks, at both micro- and macro- levels, will provide numerous sources for the development of general network theory and methodology and also facilitate the development of theory and methodology of biological networks.

Biological network analysis is a fast moving science. Many core scientific issues, for example, ecological structure, co-evolution, co-extinction and biodiversity conservation in ecology, and cancer development and metabolic regulation in health science, etc., are expected to be addressed by network approaches and network analysis. Network analysis is becoming the core methodology to treat complex biological systems. As the fast development of this area, more and more papers on biological networks are published.

At present explosive numbers of quality papers on biological networks are being published each year. The initiation of the journal, Network Biology, will provide a public and unified platform for the publication of these studies. From this integrated and unique journal, researchers, university teachers and students will thoroughly have an in-depth and complete insight on knowledge, methodology and recent advances of biological networks.

In the view of system dynamics, biological networks are always self-organized systems with emergent, autonomous and adaptive properties. Their dynamics can be represented by agent-based modeling, individual-based modeling and some other methodologies like neural network modeling. Therefore, agent-based modeling, individual-based modeling, self-organization of biological systems, and neural network modeling, etc., fall into the aims and scope of the journal, Network Biology.

The NETWORK BIOLOGY (ISSN 2220-8879) is an open access, peer-reviewed online journal that considers scientific articles in all different areas of network biology. It is the transactions of the International Society of Network Biology.It dedicates to the latest advances in network biology. The goal of this journal is to keep a record of the state-of-the-art research and promote the research work in these fast moving areas. The topics to be covered by Network Biology include, but are not limited to:

•Theories, algorithms and programs of network analysis
•Innovations and applications of biological networks
•Dynamics, optimization and control of biological networks
•Ecological networks, food webs and natural equilibrium
•Co-evolution, co-extinction, biodiversity conservation
•Metabolic networks, protein-protein interaction networks, biochemical reaction networks, gene networks, transcriptional regulatory networks, cell cycle networks, phylogenetic networks, network motifs
•Physiological networks
•Network regulation of metabolic processes, human diseases and ecological systems
•Social networks, epidemiological networks
•System complexity, self-organized systems, emergence of biological systems, agent-based modeling, individual-based modeling, neural network modeling, and other network-based modeling, etc.
We are also interested in short communications that clearly address a specific issue or completely present a new ecological network, food web, or metabolic or gene network, etc.

Authors can submit their works to the email box of this journal, networkbiology@iaees.org. All manuscripts submitted to this journal must be previously unpublished and may not be considered for publication elsewhere at any time during review period of this journal. Authors are asked to read Author Guidelines before submitting manuscripts.

In addition to free submissions from authors around the world, special issues are also accepted. The organizer of a special issue can collect submissions (yielded from a research project, a research group, etc.) on a specific research topic, or submissions of a scientific conference for publication of special issue.

Network Biology 
ISSN 2220-8879
URL: http://www.iaees.org/publications/journals/nb/online-version.asp
RSS feed: http://www.iaees.org/publications/journals/nb/rss.xml
E-mail: networkbiology@iaees.org
Editor-in-Chief: WenJun Zhang"

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov inequality is also proven.

We study the model of interacting agents proposed by Chatterjee (2003) that allows agents to both save and exchange wealth. Closed equations for the wealth distribution are developed using a mean field approximation. We show that when all agents have the same fixed savings propensity, subject to certain well defined approximations defined in the text, these equations yield the conjecture proposed by Chatterjee (2003) for the form of the stationary agent wealth distribution. If the savings propensity for the equations is chosen according to some random distribution we show further that the wealth distribution for large values of wealth displays a Pareto like power law tail, ie P (w) ∼ w1+a. However the value of a for the model is exactly 1. Exact numerical simulations for the model illustrate how, as the savings distribution function narrows to zero, the wealth distribution changes from a Pareto form to to an exponential function. Intermediate regions of wealth may be approximately d...

You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.

The recent interest in aspects common to quantum information and condensed matter has prompted a flory of activity at the border of these disciplines that were far distant untill few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermion and boson model systems. Both bipartite and multipartite entanglement will be considered. In equilibrium we show how entanglement is tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.

This paper explores the assumptions underpinning de Broglie’s concept of a wavepacket and related questions and issues. It also explores how the alternative – the ring current model of an electron (or of matter-particles in general) – relates to Louis de Broglie’s λ = h/p relation and rephrases the theory in terms of the wavefunction as well as the wave equation(s) for an electron in free space.

A la mezcla, al roce, a la colisión, a la intersección, al cruce, en fin, a la imaginación impura. [...]" J. Wagensberg 'Ideas para la imaginación impura' ---"Vosotros, los hombres, no sabéis medir vuestros días. Medís solo su longitud y decís que duran veinticuatro horas. Sin embargo, vuestros días tienen a veces una profundidad mayor que la longitud, y esta profundidad puede equivaler a un mes o incluso a un año de los días lineales. Por eso, no sabéis abarcar vuestra vida..." M. Pavic 'El último amor en Constantinopla'

Within the abstract framework of dynamical system theory we describe a general approach to the Transient (or Evans-Searles) and Steady State (or Gallavotti-Cohen) Fluctuation Theorems of non-equilibrium statistical mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. Besides its conceptual simplicity, another advantage of our approach is its natural extension to

A brief explanation of the development of the continuous concept of matter, as represented by the general theory of relativity and the unified field theory, is offered, as opposed to the discrete theory of matter represented by quantum mechanics which has been normally interpreted by the Copenhagen School. Throughout the modern development of these two concepts - the discrete and continuous aspects of nature - the thread of wave/particle duality is intimately woven. Yet they do not seem to be compatible. Wave/particle duality is a unifying concept while the debate over the discrete and continuous natures of matter is anything but unifying. In spite of the overwhelming presence of the discrete-continuity debate in physics, wave/particle duality remains important because the concept is an honest attempt to bring these two opposing views together as one universal concept. It seems strange that the physics community can allow this vast discrepancy in basic notions of nature to exist and persist, but it does so in spite of all attempts to describe nature within the context of either one or the other of these seemingly incompatible concepts. This paper was written well before the Standard Particle Model became the dominant theory of the quantum. Yet, paradoxically, parts of this argument have become ever more relevant because of the dominance of the Standard Model, which is supposedly a unifying theory, but seems to stress the differences between discrete and continuous rather than attempt to either unify them or demonstrate their physical compatibility in any manner.

We investigate the simulation of fermionic systems on a quantum computer. We show in detail how quantum computers avoid the dynamical sign problem present in classical simulations of these systems, therefore reducing a problem believed to be of exponential complexity into one of polynomial complexity. The key to our demonstration is the spin-particle connection (or generalized Jordan-Wigner transformation) that allows exact algebraic invertible mappings of operators with different statistical properties. We give an explicit implementation of a simple problem using a quantum computer based on standard qubits.

We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost-Connes naturally arises by extension of scalars from the "field with one element" to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that "field", while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. 6 1 "truc" in French 7

We study the efficiency at maximum power, η * , of engines performing finite-time Carnot cycles between a hot and a cold reservoir at temperatures T h and Tc, respectively. For engines reaching Carnot efficiency ηC = 1 − Tc/T h in the reversible limit (long cycle time, zero dissipation), we find in the limit of low dissipation that η * is bounded from above by ηC /(2 − ηC) and from below by ηC /2. These bounds are reached when the ratio of the dissipation during the cold and hot isothermal phases tend respectively to zero or infinity. For symmetric dissipation (ratio one) the Curzon-Ahlborn efficiency ηCA = 1 − Tc/T h is recovered.

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior, such as Gutenberg-Richter scaling and the relation between large and small events, which is the basis for various forecasting methods. Although cellular automaton models have been studied extensively in the long-range stress transfer limit, this limit has not been studied for the Burridge-Knopoff model, which includes more realistic friction forces and inertia. We find that the latter model with long-range stress transfer exhibits qualitatively different behavior than both the long-range cellular automaton models and the usual Burridge-Knopoff model with nearest neighbor springs, depending on the nature of the velocity-weakening friction force. This results have important implications for our understanding of earthquakes and other driven dissipative systems. PACS numbers: 91.30.Px, 02.60.Cb,05.20.-y, 05.45.-a Earthquake faults are examples of driven dissipative systems [1]

We discuss a Statistical Mechanics approach in the manner of Edwards to the "inherent states" (defined as the stable configurations in the potential energy landscape) of glassy systems and granular materials. We show that at stationarity the inherent states are distributed according a generalized Gibbs measure obtained assuming the validity of the principle of maximum entropy, under suitable constraints. In particular we consider three lattice models (a diluted Spin Glass, a monodisperse hard-sphere system under gravity and a hard-sphere binary mixture under gravity) undergoing a schematic "tap dynamics", showing via Monte Carlo calculations that the time average of macroscopic quantities over the tap dynamics and over such a generalized distribution coincide. We also discuss about the general validity of this approach to non thermal systems.

Much attention has been devoted to predicting and controlling fracture toughness in ferrous weldments. Extensive studies relating toughness with welding parameters and weldment composition have been made. All such studies have sought to define the important variables affecting toughness. Unfortunately, such studies often resulted in linear regressions of one variable against toughness, fracture mode was not held constant, and no efforts were made to relate the variable investigated to quantitative features of the weldment microstructure.

In generalized statistical mechanics the second link is customarily relaxed. Of course, the generalized exponential function defining the probability distribution function after inversion, produces a generalized logarithm Λ(pi). But, in general, the mean value of −Λ(pi) is not the entropy of the system. Here we reconsider the question first posed in [Phys. Rev. E 66, 056125 (2002) and 72, 036108 ], if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics.

Boltzmann's 1872 derivation of the H-theorem was of great significance
because it provided a basis for the second law of thermodynamics in terms of the
molecular/kinetic theory of heat. By showing that a statistical treatment of the
many molecules comprising a gas can produce a monotonic decrease of an
entropy-like quantity, H ~ -S , he provided the essential insight into the
connection between the second law and the evolution of systems through
macroscopic states occupying progressively larger volumes in phase space.
    However, it is a common misconception that an analysis like that given by
Boltzmann demonstrates that the second law of thermodynamics would be
observed in a universe of particles whose motions are completely described by
the laws of classical mechanics. I attempt to clarify that this is a misconception
by showing that an element introduced into Boltzmann's derivation as simply an
approximation to the dynamics expected under classical mechanics, in fact
introduces a new feature into the dynamics of the model system. It is shown
that this added feature, present in the model but not in a classical mechanical
universe, is solely responsible for the monotonic behavior of H. Hence, while
this type of analysis provides an understanding of how the second law comes
about, it does not stay within the confines of classical mechanics in doing so.
Thus it is not a derivation of the second law of thermodynamics just from the
laws of classical mechanics for a system with many degrees of freedom and a
low-entropy initial condition. The implications of this conclusion are important
for our understanding of the physical basis of the second law of
thermodynamics.

This article reports an open discussion that took place during the Keenan Symposium "Meeting the EntropyChallenge" (held in Cambridge, Massachusetts, on October 4, 2007) following the short presentations-each reported as aseparate article in the present volume-byAdrian

Characterizing and quantifying quantum correlations in states of many-particle systems is at the core of a full understanding of phase transitions in matter. In this work, we continue our investigation of the notion of generalized entanglement [Barnum et al., Phys. Rev. A 68, 032308 (2003)] by focusing on a simple Liealgebraic measure of purity of a quantum state relative to an observable set. For the algebra of local observables on multi-qubit systems, the resulting local purity measure is equivalent to a recently introduced global entanglement measure [Meyer and Wallach, J. Math. Phys. 43, 4273 (2002)]. In the condensed-matter setting, the notion of Lie-algebraic purity is exploited to identify and characterize the quantum phase transitions present in two exactly solvable models, namely the Lipkin-Meshkov-Glick model and the spin-1 2 anisotropic XY model in a transverse magnetic field. For the latter, we argue that a natural fermionic observable set arising after the Jordan-Wigner transformation better characterizes the transition than alternative measures based on qubits. This illustrates the usefulness of going beyond the standard subsystem-based framework while providing a global disorder parameter for this model. Our results show how generalized entanglement leads to useful tools for distinguishing between the ordered and disordered phases in the case of broken symmetry quantum phase transitions. Additional implications and possible extensions of concepts to other systems of interest in condensed-matter physics are also discussed.

A theoretically based closed-form analytical equation for the radial distribution function, g͑r͒, of a fluid of hard spheres is presented and used to obtain an accurate analytic representation. The method makes use of an analytic expression for the short-and long-range behaviors of g͑r͒, both obtained from the Percus-Yevick equation, in combination with the thermodynamic consistency constraint. Physical arguments then leave only three parameters in the equation of g͑r͒ that are to be solved numerically, whereas all remaining ones are taken from the analytical solution of the Percus-Yevick equation.

The mixing properties (or sensitivity to initial conditions) of the two-dimensional Henon map have been explored numerically at the edge of chaos. Three independent methods, which have been developed and used so far for one-dimensional maps, have been used to accomplish this task. These methods are (i) the measure of the divergence of initially nearby orbits, (ii) analysis of the multifractal spectrum, and (iii) computation of nonextensive entropy increase rates. The results obtained closely agree with those of the one-dimensional cases and constitute a verification of this scenario in two-dimensional maps. This obviously makes the idea of weak chaos even more robust.

Biological cells sense external chemical stimuli in their environment using cell-surface receptors. To increase the sensitivity of sensing, receptors often cluster, most noticeably in bacterial chemotaxis, a paradigm for signaling and sensing in general. While amplification of weak stimuli is useful in absence of noise, its usefulness is less clear in presence of extrinsic input noise and intrinsic signaling noise. Here, exemplified on bacterial chemotaxis, we combine the allosteric Monod-Wyman- Changeux model for signal amplification by receptor complexes with calculations of noise to study their interconnectedness. Importantly, we calculate the signal-to-noise ratio, describing the balance of beneficial and detrimental effects of clustering for the cell. Interestingly, we find that there is no advantage for the cell to build receptor complexes for noisy input stimuli in absence of intrinsic signaling noise. However, with intrinsic noise, an optimal complex size arises in line with estimates of the sizes of chemoreceptor complexes in bacteria and protein aggregates in lipid rafts of eukaryotic cells.

The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated system&#39;s Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.

The Schwinger boson mean field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long range order in the ground state. We observe two branches of the low lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T = 0K. These results agree with the established numerical results quite well. At finite temperatures, the long range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T > 0K, are calculated. The T χ uni has a minimum at intermediate temperatures and the spin correlation length behaves as T -1 at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures.

The collisionless Boltzmann equation is generalized herein using the Green function theory proposed recently by A.K. Rajagopal et al. [Phys. Rev. Lett. 80, 3911 (1998)] to describe nonextensive systems based on Tsallis formalism. Its invariance with the nonextensive index, q, and some conservative laws useful in the description of sound propagation are verified. In this context, the Wigner distribution has also been introduced. Furthermore, the formalism developed in the research is applied for fermionic system, to obtain the dynamic dielectric response function.

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

We derive a 1/c-expansion for the single-particle density matrix of a strongly interacting timedependent one-dimensional Bose gas, described by the Lieb-Liniger model (c denotes the strength of the interaction). The formalism is derived by expanding Gaudin's Fermi-Bose mapping operator up to 1/c-terms. We derive an efficient numerical algorithm for calculating the density matrix for time-dependent states in the strong coupling limit, which evolve from a family of initial conditions in the absence of an external potential. We have applied the formalism to study contraction dynamics of a localized wave packet upon which a parabolic phase is imprinted initially.

Fluctuations of the instantaneous local Lagrangian strain $\epsilon_{ij}(\bf{r},t)$, measured with respect to a static ``reference'' lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse- grained by averaging them within subsystems (of size $L_b$) of a system (of total size $L$) in the canonical ensemble. Using a simple finite size scaling theory we predict the behaviour of the fluctuations $<\epsilon_{ij}\epsilon_{kl}>$ as a function of $L_b/L$ and extract elastic constants of the system {\em in the thermodynamic limit} at nonzero temperature. Our method is simple to implement, efficient and general enough to be able to handle a wide class of model systems including those with singular potentials without any essential modification. We illustrate the technique by computing isothermal elastic constants of the ``soft'' and the hard disk triangular solids in two dimensions from molecular dynamics and Monte Carlo simulations. We compare our results with those from earlier simulations and density functional theory.

The Boltzmann-Gibbs-von Neumann entropy of a large part (of linear size L) of some (much larger) d-dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to L d−1 . Here we show, for d = 1, 2, that the (nonadditive) entropy Sq satisfies, for a special value of q = 1, the classical thermodynamical prescription for the entropy to be extensive, i.e., Sq ∝ L d . Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited, by M. Gell-Mann, Y. Sato and one of us (C.T.), in mathematical models with scale-invariant correlations. Finally, we find that the system critical features are marked by a maximum of the special entropic index q.

We study the model of interacting agents proposed by Chatterjee (2003) that allows agents to both save and exchange wealth. Closed equations for the wealth distribution are developed using a mean field approximation. We show that when all agents have the same fixed savings propensity, subject to certain well defined approximations defined in the text, these equations yield the conjecture proposed by Chatterjee (2003) for the form of the stationary agent wealth distribution. If the savings propensity for the equations is chosen according to some random distribution we show further that the wealth distribution for large values of wealth displays a Pareto like power law tail, ie P (w) ∼ w 1+a. However the value of a for the model is exactly 1. Exact numerical simulations for the model illustrate how, as the savings distribution function narrows to zero, the wealth distribution changes from a Pareto form to to an exponential function. Intermediate regions of wealth may be approximately described by a power law with a > 1. However the value never reaches values of ∼ 1.6 − 1.7 that characterise empirical wealth data. This conclusion is not changed if three body agent exchange processes are allowed. We conclude that other mechanisms are required if the model is to agree with empirical wealth data.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

We study the quantum phase diagram and excitation spectrum of the frustrated J1-J2 spin-1/2 Heisenberg Hamiltonian. A hierarchical mean-field approach, at the heart of which lies the idea of identifying relevant degrees of freedom, is developed. Thus, by performing educated, manifestly symmetry preserving mean-field approximations, we unveil fundamental properties of the system. We then compare various coverings of the square lattice with plaquettes, dimers and other degrees of freedom, and show that only the symmetric plaquette covering, which reproduces the original Bravais lattice, leads to the known phase diagram. The intermediate quantum paramagnetic phase is shown to be a (singlet) plaquette crystal, connected with the neighbouring Néel phase by a continuous phase transition. We also introduce fluctuations around the hierarchical mean-field solutions, and demonstrate that in the paramagnetic phase the ground and first excited states are separated by a finite gap, which closes in the Néel and columnar phases. Our results suggest that the quantum phase transition between Néel and paramagnetic phases can be properly described within the Ginzburg-Landau-Wilson paradigm.