thermostats for slow configurational modes thermostats are dynamical equations used to model thermodynamic variables such as temperature and pressure in molecular simulations for computationally intensive problems such as the simulation of biomolecules we propose to average over fast mo mentum degrees of freedom and construct thermostat equations in configuration space the equations of motion are deterministic analogues of the smoluchowski dynamics in the method of stochastic differential equations

non ergodicity of the nose hoover thermostatted harmonic oscillator the nose hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical gibbs distribution newton s equations are modified by coupling an additional reservoir variable to the physical variables the correct sampling of the phase space according to the gibbs measure is dependent on the nose hoover dynamics being ergodic hoover presented numerical experiments to show that the nose hoover dynamics are non ergodic when applied to the harmonic oscillator in this article we prove that the nose hoover thermostat does not give an ergodynamical system for the one dimensional harmonic oscillator when the mass of the reservoir is large our proof of non ergodicity uses kam theory to demonstrate the existence of invariant tori for the nose hoover dynamical system that separate phase space into invariant regions we present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass

generalized ensemble and tempering simulations a unified view from the underlying master equations we derive one dimensional stochastic processes that describe generalized ensemble simulations as well as tempering simulated and parallel simulations the representations obtained are either in the form of a one dimensional fokker planck equation or a hopping process on a one dimensional chain in particular we discuss the conditions under which these representations are valid approximate markovian descriptions of the random walk in order parameter or control parameter space they allow a unified discussion of the stationary distribution on as well as of the stationary flow across each space we demonstrate that optimizing the flow is equivalent to minimizing the first passage time for crossing the space and discuss the consequences of our results for optimizing simulations finally we point out the limitations of these representations under conditions of broken ergodicity

superstatistics thermodynamics and fluctuations a thermodynamiclike formalism is developed for superstatistical systems based on conditional entropies this theory takes into account large scale variations of intensive variables of systems in nonequilibrium stationary states ordinary thermodynamics is recovered as a special case of the present theory and corrections to it can systematically be evaluated a generalization of einstein s relation for fluctuations is presented using a maximum entropy condition

construction of an extended invariant for an arbitrary ordinary differential equation with its development in a numerical integration algorithm for an arbitrary ordinary differential equation ode a scheme for constructing an extended ode endowed with a time invariant function is here proposed this scheme enables us to examine the accuracy of the numerical integration of an ode that may itself have had no invariant these quantities are constructed by referring to the nose hoover molecular dynamics equation and its related conserved quantity by applying this procedure to several molecular dynamics equations the conventional conserved quantity individually defined in each dynamics can be reproduced in a uniform generalized way our concept allows a transparent outlook underlying these quantities and ideas developing the technique for a certain class of odes we construct a numerical integrator that is not only explicit and symmetric but preserves a unit jacobian for a suitably defined extended ode which also provides an invariant our concept is thus to simply build a divergence free extended ode whose solution is just a lift up of the original ode and to constitute an efficient integrator that preserves the phase space volume on the extended system we present precise discussions about the general mathematical properties of the integrator and provide specific conditions that should be incorporated for practical applications

generalized entropies and the transformation group of superstatistics superstatistics describes statistical systems that behave like superpositions of different inverse temperatures so that the probability distribution is where the kernel f is nonnegative and normalized f d 1 we discuss the relation between this distribution and the generalized entropic form the first three shannon khinchin axioms are assumed to hold it then turns out that for a given distribution there are two different ways to construct the entropy one approach uses escort probabilities and the other does not the question of which to use must be decided empirically the two approaches are related by a duality the thermodynamic properties of the system can be quite different for the two approaches in that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes the transformation group is the euclidean group in one dimension

the theory of ℛ is undecidable the three quantifier theory of r t the recursively enumerable degrees under turing reducibility was proven undecidable by lempp nies and slaman 1998 the two quantifier theory includes the lattice embedding problem and its decidability is a long standing open question a negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational we prove the undecidability of a fragment of the theory of r that lies between the two and three quantifier theories with t but includes function symbols theorem the two quantifier theory of r v the r e degrees with turing reducibility supremum and infimum taken to be any total function extending the infimum relation on r is undecidable the same result holds for various lattices of ideals of r which are natural extensions of r preserving join and infimum when it exits

miniversal deformations of marked matrices abstract given the set of square matrices m m n m c that keep the subspace w c n x 0 c n m invariant we obtain the implicit form of a miniversal deformation of a matrix a m and we compute it explicitely when this matrix is marked this is if there is a permutation matrix p m n m c such that p 1 ap is a jordan matrix we derive some applications to tackle the classical carlson problem

tsallis dynamics using the nose hoover approach on the basis of the nos e hoover method we developed a deterministic algorithm that produces an arbitrary probability density an ordinary differential equation in the algorithm can realize the tsallis distribution density the tsallis distribution has been considered a candidate of a distribution that represents a physical system in a heat bath the tsallis distribution density employed in this algorithm is defined using a full energy function form e x p along with the tsallis index qg 1 using the current equation numerical simulations were performed for simple systems and the tsallis distributions were observed

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

double density dynamics realizing a joint distribution of a physical system and a parameter system to perform a variety of types of molecular dynamics simulations we created a deterministic method termed double density dynamics ddd which realizes an arbitrary distribution for both physical variables and their associated parameters simultaneously specifically we constructed an ordinary differential equation that has an invariant density relating to a joint distribution of the physical system and the parameter system a generalized density function leads to a physical system that develops under nonequilibrium environment describing superstatistics the joint distribution density of the physical system and the parameter system appears as the radon nikodym derivative of a distribution that is created by a scaled long time average generated from the flow of the differential equation under an ergodic assumption the general mathematical framework is fully discussed to address the theoretical possibility of our method and a numerical example representing a 1d harmonic oscillator is provided to validate the method being applied to the temperature parameters

symmetric explicit numerical integrator for molecular dynamics equations of motion with a generalized friction a general mathematical scheme to construct symmetric explicit numerical integrators of newtonian equations of motion endowed with a generalized friction is provided for a molecular dynamics md study the exact integrations are done for all the decomposed vector fields including the one that contains the friction term on the basis of the symmetric composition scheme with the adjoint for the resulting maps integrators with any local order of accuracy can be systematically constructed among them the second order p2s1 integrator gives the least evaluation of atomic force and potential which are most time consuming in md simulations as examples of the friction function three functional types are considered constant laurent polynomial and exponential with respect to the kinetic energy several md equations of motion fall into these categories and the numerical examinations of their integrators using model systems give positive results on the accuracy and efficiency the extended phase space scheme also presents an invariant function which allows us to easily detect numerical errors in the integration process by monitoring the function value

molecular dynamics sampling scheme realizing multiple distributions we present a molecular dynamics sampling scheme in which any summation of multiple arbitrarily given distributions can be realized deterministically by a certain theoretical guide our scheme thus provides a static view of the composition of the distributions rather than a dynamic view in which some parameter is switched at a certain time in the simulation process the proposed method induces the orbit to jump automatically among different phase space regions without the use of any artificial timing parameter in addition the proposed method does not require an assumption that the density of states is smooth we applied it to multiple tsallis distributions and established a suitable series of parameter values for which the sum of the distributions allows broad sampling numerical simulations applied to fundamental models with multi tsallis distributions showed efficient sampling characterized by an energy trajectory that was totally different from that associated with each single distribution

extensions of jordan bases for invariant subspaces of a matrix a characterization is obtained for the matrices a with the property that every some jordan basis of every a invariant subspace can be extended to a jordan basis of a these results are based on a criterion for a jordan basis of an invariant subspace to be extendable to a jordan basis of the whole space the criterion involves two concepts the constancy property and the depth property

computational processes observers and turing incompleteness we propose a formal definition of wolfram s notion of computational processes based on iterated transducers together with a weak observer a model of computation that captures some aspects of physics like computation these processes admit a natural classification into decidable intermediate and complete where intermediate processes correspond to recursively enumerable sets of intermediate degree in the classical setting it is shown that a standard finite injury priority argument will not suffice to establish the existence of an intermediate computational process

on q gaussians and exchangeability the q gaussian distributions introduced by tsallis are discussed from the point of view of variance mixtures of normals and exchangeability for each there is a q gaussian distribution that maximizes the tsallis entropy under suitable constraints this paper shows that q gaussian random variables can be represented as variance mixtures of normals when q 1 these variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables thereby providing a possible model that has been extensively studied in probability theory the formulation provided has the additional advantage of yielding for each q a process which is naturally the q analog of the brownian motion explicit mixing distributions for q gaussians should facilitate applications to areas such as option pricing the model might provide insight into the study of superstatistics

the π theory of the computably enumerable turing degrees is undecidable we show the undecidability of the 3 theory of the partial order of enumerable turing degrees 0 introduction recursively enumerable henceforth called enumerable sets arise naturally in many areas of mathematics for instance in the study of elementary theories as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups putting the enumerable sets into context with each other in various ways yields structures whose study has for long been a mainstay of computability theory if the sets are related in the most elementary way namely by inclusion one obtains a distributive lattice e with very complex algebraic properties another way to compare sets is to look at the information content turing reducibility is a very general but the most widely accepted concept of relative computability a setx of natural numbers is turing reducible to y iff the answer to n x can be determined by a turing machine computation which can use answers to oracle questions y y during the computation for more restricted notions of relative computability one would for instance place a priori 1991 mathematics subject classification 03d25 03d35

complex networks emerging from fluctuating random graphs analytic formula for the hidden variable distribution in analogy to superstatistics which connects boltzmann gibbs statistical mechanics to its generalizations through temperature fluctuations complex networks are constructed from fluctuating erd os r enyi random graphs using a quantum mechanical method the exact analytic formula for the hidden variable distribution is presented which describes the nature of the fluctuations and generates a generic degree distribution through the poisson transformation as an example a static scale free network is discussed and the corresponding hidden variable distribution is found to decay as a power law and to diverge at the origin

superstatistics in nanoscale electrochemical systems stochastic electrochemical reaction steps on nanosized electrodes are non markovian when externally driven by an applied voltage we show that compared to the markovian case when external driving is absent nanoscale electrochemical systems obey a superstatistics characterized by a superposition of tsallis q indices the distribution of tsallis q indices along stochastic trajectories can be calculated from the electrochemical master equation and normal distributions from boltzmann gibbs thermostatistics are recovered in the thermodynamic limit the infinite electrode size limit although on the nanoscale the external control makes intricate correlations between the microstates in the superstatistical frame one can still address the microstates as if they were uncorrelated the resulting superstatistical entropic form is additive in this frame and tsallis indices have on the time average values which is indeed an example of a superstatistical system where no ad hoc distribution has to be assumed for the fluctuations rather the distribution is directly calculated from a mesoscopic master equation without freely adjustable parameters

superstatistical generalizations of wishart laguerre ensembles of random matrices using beck and cohen s superstatistics we introduce in a systematic way a family of generalised wishart laguerre ensembles of random matrices with dyson index 1 2 and 4 the entries of the data matrix are gaussian random variables whose variancesfluctuate from one sample to another according to a certain probability density f and a single deformation parameter three superstatistical classes for f are usually considered 2 inverse 2 and log normal distributions while the first class already considered by two of the authors leads to a power law decay of the spectral density we here introduce and solve exactly a superposition of wishart laguerre ensembles with inverse 2 distribution the corresponding macroscopic spectral density is given by a deformation of the semi circle and marcenko pastur laws on a non compact support with exponential tails after discussing in detail the validity of wigner s surmise in the wishart laguerre class we introduce a generalised dependent surmise with stretched exponential tails which well approximates the individual level spacing distribution in the bulk the analytical results are in excellent agreement with numerical simulations to illustrate our findings we compare the 2 and inverse 2 class to empirical data from financial covariance matrices

coupled nose hoover equations of motions without time scaling the nose hoover nh equation of motion is widely used in molecular dynamics simulations it enables us to set a constant temperature and produce the canonical distribution for a target physical system for the purpose of investigating the physical system under fluctuating temperature we have introduced a coupled nose hoover equation in our previous work fukuda and moritsugu 2015 j phys a math theor 48 455001 the coupled nh equation implements a fluctuating heat bath temperature in the nh equation of the physical system and also keeps a statistically complete description via an invariant measure of the total system composed of the physical system and a temperature system however a difficulty lies in that the time development of the physical system may not correspond to the realistic physical process because of the need of a scaled time average to compute thermodynamical quantities the current work gives a solution by presenting a new scheme which is free from the scaled time but retains the statistical description by use of simple model systems we validate the current scheme and compare with the original scheme the sampling property of the current scheme is also clarified to investigate the effect of function setting used for the distribution of the total system

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators