the representation of integers as sums of squares we present a uniform method by which we obtain an explicit formula for the number of representations of an integer as the sum of n squares for each n in the range 2 n 8 the final formulas for odd n are new though those for even n are not we also present a formula for the dominant term for the number for an arbitrary n introduction in this paper we are interested in the sum of squares x n

microcanonical quantum fluctuation theorems previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to hamiltonians with degenerate spectra in the particular case of microcanonical initial states explicit expressions for the characteristic function and the corresponding probability density of work are formulated their classical limit as well as their relations to the corresponding canonical expressions are discussed a fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final hamiltonians from this crooks type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time reversed process this entropy from work theorem provides an experimentally accessible way to measure entropies

quantum fluctuation relations for the lindblad master equation an open quantum system interacting with its environment can be modeled under suitable assumptions as a markov process described by a lindblad master equation in this work we derive a general set of fluctuation relations for systems governed by a lindblad equation these identities provide quantum versions of jarzynski hatano sasa and crooks relations in the linear response regime these fluctuation relations yield a fluctuation dissipation theorem fdt valid for a stationary state arbitrarily far from equilibrium for a closed system this fdt reduces to the celebrated callen welton kubo formula

residuated skew lattices abstract skew lattices are one of the most successful non commutative generalizations of lattices motivated by the study of residuation on ordered structures and that of skew boolean algebras and skew heyting algebras in this paper we introduce the concept of residuated skew lattices as a new non commutative version of residuated lattices the axiomatization and localization for residuated skew lattices are obtained moreover a characterization of residuation on skew lattices via adjointness is presented meanwhile three special subvarieties distributive residuated skew lattices skew mtl algebras and skew bl algebras are investigated in particular we show that every skew heyting algebra is a reduct of a special residuated skew lattice

quantum work statistics and resource theories bridging the gap through renyi divergences the work performed on or extracted from a nonautonomous quantum system described by means of a two point projective measurement approach is a stochastic variable we show that the cumulant generating function of work can be recast in the form of quantum r enyi ensuremath alpha divergences and by exploiting the convexity of this cumulant generating function derive a single parameter family of bounds for the first moment of work higher order moments of work can also be obtained from this result in this way we establish a link between quantum work statistics in stochastic approaches and resource theories for quantum thermodynamics a theory in which r enyi ensuremath alpha divergences take a central role to explore this connection further we consider an extended framework involving a control switch and an auxiliary battery which is instrumental to reconstructing the work statistics of the system we compare and discuss our bounds on the work distribution to findings on deterministic work studied in resource theoretic settings

stiffness of probability distributions of work and jarzynski relation for non gibbsian initial states we consider closed quantum systems into which baths may be integrated that are driven i e subject to time dependent hamiltonians our point of departure is the assumption that if systems start in non gibbsian states at some initial energies the resulting probability distributions of work may be largely independent of the specific initial energies it is demonstrated that this assumption has some far reaching consequences e g it implies the validity of the jarzynski relation for a large class of non gibbsian initial states by performing numerical analysis on integrable and nonintegrable spin systems we find the above assumption fulfilled for all examples considered through an analysis based on fermi s golden rule we partially relate these findings to the applicability of the eigenstate thermalization ansatz to the respective driving operators

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

test of fluctuation theorems in non markovian open quantum systems we study fluctuation theorems for open quantum systems with a non markovian heat bath using the approach of quantum master equations and examine the physical quantities that appear in those fluctuation theorems the approach of markovian quantum master equations to the fluctuation theorems was developed by esposito and mukamel phys rev e 73 046129 2006 we show that their discussion can be formally generalized to the case of a non markovian heat bath when the local system is linearly connected to a gaussian heat bath with the spectrum distribution of the drude form we found by numerically simulating the spin boson model in non markovian regime that the detailed balance condition is well satisfied except in a strongly nonequilibrium transient situation and hence our generalization of the definition of the entropy production is almost always legitimate therefore our generalization of the fluctuation theorem seems meaningful in wide regions

thermodynamic cost of acquiring information connections between information theory and thermodynamics have proven to be very useful to establish bounding limits for physical processes ideas such as landauer s erasure principle and information assisted work extraction have greatly contributed not only to broadening our understanding about the fundamental limits imposed by nature but also paving the way for practical implementations of information processing devices the intricate information thermodynamics relation also entails a fundamental limit on parameter estimation establishing a thermodynamic cost for information acquisition we show that the amount of information that can be encoded in a physical system by means of a unitary process is limited by the dissipated work during the implementation of the process this includes a thermodynamic tradeoff for information acquisition likewise the information acquisition process is ultimately limited by the second law of thermodynamics this tradeoff for information acquisition may find applications in several areas of knowledge

the tasaki crooks quantum fluctuation theorem starting out from the recently established quantum correlation function expression of the characteristic function for the work performed by a force protocol on the system in talkner et al 2007 phys rev e 75 050102 preprint cond mat 0703213 the quantum version of the crooks fluctuation theorem is shown to emerge almost immediately by the mere application of an inverse fourier transformation

fluctuation theorems work is not an observable 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 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 lattice theoretic structure of the sets of triangular norms and semi copulas abstract the lattice theoretic structure of the set of semi copulas and some of its subsets is investigated as a relevant case we obtain that the set of commutative semi copulas is a lattice completion of the set of triangular norms

lag inequality for birth death processes with time dependent rates a fluctuation theorem has recently been derived for systems described by univariate birth death equations or chemical master equations seifert u 2004 j phys a math gen 37 l517 21 we discuss an inequality that follows from this theorem focusing on the case of poissonian stationary distributions we show that this inequality can also be derived from a deterministic rate equation in the limit of large system size we relate these results to recent work on transitions between nonequilibrium stationary states

occurrence of discontinuities in the performance of finite time quantum otto cycles we study a quantum otto cycle in which the strokes are performed in finite time the cycle involves energy measurements at the end of each stroke to allow for the respective determination of work we then optimize for the work and efficiency of the cycle by varying the time spent in the different strokes and find that the optimal value of the ratio of time spent on each stroke goes through sudden changes as the parameters of this cycle vary continuously the position of these discontinuities depends on the optimized quantity under consideration such as the net work output or the efficiency

fluctuation theorem for birth death or chemical master equations with time dependent rates for systems described by univariate birth death or chemical master equations driven out of equilibrium by externally controlled time dependent transition rates a nonlinear fluctuation theorem is derived for paradigmatic chemical reactions this theorem acquires a particularly transparent form

filters of residuated lattices and triangle algebras an important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic is that of a filter filters can be used amongst others to define congruence relations specific kinds of filters include boolean filters and prime filters in this paper we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections triangle algebras characterize interval valued residuated lattices

temporal disorder and fluctuation theorem in chemical reactions we report the analytical study of a class of chemical reactions described as birth and death stochastic processes ruled by a master equation compatible with the mass action law of chemical kinetics we solve analytically this master equation to find the generating functions of the fluctuating fluxes and of the lebowitz spohn action functional these generating functions are explicitly shown to obey fluctuation theorems in the case of fluxes we derive relations for the nonlinear response coefficients extending onsager s reciprocity relations moreover symmetry relations reminiscent of the fluctuation theorem are obtained for the finite time probability distributions of the fluxes the temporal disorder of the stochastic process is also characterized and related to the thermodynamic entropy production

heat transfer operators associated with quantum operations any quantum operation applied on a physical system is performed as a unitary transformation on a larger extended system if the extension used is a heat bath in thermal equilibrium the concomitant change in the state of the bath necessarily implies a heat exchange with it the dependence of the average heat transferred to the bath on the initial state of the system can then be found from the expectation value of a hermitian operator which is named as the heat transfer operator hto the purpose of this paper is to investigate the relation between the htos and the associated quantum operations since any given quantum operation on a system can be realized by different baths and unitaries many different htos are possible for each quantum operation on the other hand there are also strong restrictions on the htos which arise from the unitarity of the transformations the most important of these is the landauer erasure principle this paper is concerned with the question of finding a complete set of restrictions on the htos that are associated with a given quantum operation an answer to this question has been found only for a subset of quantum operations for erasure operations these characterizations are equivalent to the generalized landauer erasure principle for the case of generic quantum operations however it appears that the htos obey further restrictions which cannot be obtained from the entropic restrictions of the generalized landauer erasure principle

quadratic diophantine equations and orders in quaternion algebras we first reformulate the main theorem on quadratic diophantine equations given in our recent book in terms of the even clifford group in the quaternary case the group is the multiplicative group of a quaternion algebra and the reformulated theorem connects a class of primitive solutions of a quadratic equation with a class of an order in the algebra in the indefinite case each class is essentially a class of binary hermitian forms in the definite case the result is an analogue of the result of gauss on the sums of three squares

skew residuated lattices we replace the so called adjointness in the definition of residuated lattice by its strict version where inequalities are replaced by equalities we prove that such structures called skew residuated lattices can be characterized as lattices with certain involutions in principal filters since skew residuated lattices have the cancellation property they are close to divisibility loops introduced by b bosbach in 1988 we show under what condition can skew residuated lattices be represented by such loops

continuous time random walk for open systems fluctuation theorems and counting statistics we consider continuous time random walks ctrw for open systems that exchange energy and matter with multiple reservoirs each waiting time distribution wtd for times between steps is characterized by a positive parameter alpha which is set to alpha 1 if it decays at least as fast as t 2 at long times and therefore has a finite first moment a wtd with alpha 1 decays as t alpha 1 a fluctuation theorem for the trajectory quantity r defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory holds for any ctrw however r can be identified as a trajectory entropy change only if the wtds have alpha 1 and satisfy separability also called direction time independence for nonseparable wtds with alpha 1 r can only be identified as a trajectory entropy change at long times and a fluctuation theorem for the entropy change then only holds at long times for wtds with 0 alpha 1 no meaningful fluctuation theorem can be derived we also show that the experimentally accessible nth moments of the energy and matter transfers between the system and a given reservoir grow as t nalpha at long times

initial ensemble dependence of jarzynski equality in the thermodynamic limit we investigate the initial ensemble dependence of the jarzynski relation for nonequilibrium work performed on a classical system in the thermodynamic limit it is shown that the jarzynski equality holds also for a microcanonical ensemble only if the work process is quasi statically slow if work is performed by a rapid perturbation the average of exponentiated work expanded on a microcanonical system markedly deviates from the jarzynski equality such deviation in fact quantifies the ratio of the energy distributions of nonequilibrium states in relative to equilibrium distribution we exemplify these points in a simple model system both for the cases of deterministic and stochastic dynamics

calculating work in adiabatic two level quantum markovian master equations a characteristic function method we present a characteristic function method to calculate the probability density functions of the inclusive work in adiabatic two level quantum markovian master equations these systems are steered by some slowly varying parameters and the dissipations may depend on time our theory is based on the interpretation of the quantum jump for the master equations in addition to the calculation we also find that the fluctuation properties of the work can be described by the symmetry of the characteristic functions which is exactly the same as in the case of isolated systems a periodically driven two level model is used to demonstrate the method

integral fluctuation theorem for the housekeeping heat the housekeeping heat qhk is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states by analysing the evolution of its probability distribution we prove an integral fluctuation theorem exp qhk 1 valid for arbitrary driven transitions between steady states we discuss gaussian limiting cases and the difference between the new theorem and both the hatano sasa and the jarzynski relation

on non associative generalizations of mv algebras and lattice ordered commutative loops the aim of the paper is to describe the connections between lattice ordered commutative loops and certain basic algebras which are a non associative generalization of mv algebras and are related to commutative semicopulas this extends the well known equivalence between lattice ordered abelian groups with strong order unit and mv algebras

integral quantum fluctuation theorems under measurement and feedback control we derive integral quantum fluctuation theorems and quantum jarzynski equalities for a feedback controlled system and a memory which registers outcomes of the measurement the obtained equalities involve the information content which reflects the information exchange between the system and the memory and take into account the back action of a general measurement contrary to the classical case the generalized second law of thermodynamics under measurement and feedback control is reproduced from these equalities

inhomogeneous quadratic forms and triangular numbers we prove an explicit formula for the number of representations of an integer as the sum of n triangular numbers for each n in the range 2 n 8 as special cases of a more general formula applicable to an inhomogeneous quadratic form over a totally real number field the formula can be derived by calculating explicitly the fourier coefficients of a certain eisenstein series that appears in the siegel weil formula for an inhomogeneous quadratic form

equivalence of two bochkov kuzovlev equalities in quantum two level systems we present two kinds of bochkov kuzovlev work equalities in a two level system that is described by a quantum markovian master equation one is based on multiple time correlation functions and the other is based on the quantum trajectory viewpoint we show that these two equalities are indeed equivalent importantly this equivalence provides us a way to calculate the probability density function of the quantum work by solving the evolution equation for its characteristic function we use a numerical model to verify these results

nonequilibrium green s function s approach to the calculation of work statistics the calculation of work distributions in a quantum many body system is of significant importance and also of formidable difficulty in the field of nonequilibrium quantum statistical mechanics to solve this problem inspired by the schwinger keldysh formalism we propose the contour integral formulation for work statistics based on this contour integral we show how to do the perturbation expansion of the characteristic function of work cfw and obtain the approximate expression of the cfw to the second order of the work parameter for an arbitrary system under a perturbative protocol we also demonstrate the validity of fluctuation theorems by utilizing the kubo martin schwinger condition finally we use noninteracting identical particles in a forced harmonic potential as an example to demonstrate the powerfulness of our approach

probability of stochastic processes and spacetime geometry abstract we made a first attempt to associate a probabilistic description of stochastic processes like birth death processes with spacetime geometry in the schwarzschild metrics on distance scales from the macro to the micro domains we idealize an ergodic system in which system states communicate through a curved path composed of transition arrows where each arrow corresponds to a positive analogous birth or death rate

states on commutative basic algebras the paper deals with states on commutative basic algebras that are a non associative generalization of mv algebras or in other words the algebraic semantics for a fuzzy logic which generalizes the lukasiewicz logic in that the conjunction is not associative states are defined in the same way as mundici s states on mv algebras as normalized finitely additive 0 1 valued functions and some results analogous to the results that are known from mv algebras are proved

statistics of work performed on a forced quantum oscillator various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator in this special case two parameters are introduced that are sufficient to completely characterize the force protocol explicit results for the characteristic function of work and the corresponding probability distribution are provided and discussed for three different types of initial states of the oscillator microcanonical canonical and coherent states depending on the choice of the initial state the probability distributions of the performed work may greatly differ this result in particular also holds true for identical force protocols general fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed

results on equality algebras in this paper by considering the notion of equality algebra which is introduced by jenei in 18 as a possible algebraic semantic for fuzzy type theory we study and show that there are relations among equality algebras and some of other logical algebras such as residuated lattice mtl algebra bl algebra mv algebra hertz algebra heyting algebra boolean algebra eq algebra and hoop algebra the aim of this paper is to find that under which conditions equality algebras are equivalent to these logical algebras

strong approximation for certain quadric fibrations with compact fibers abstract by developing the representation theory of quadratic diophantine equations we prove strong approximation with brauer manin obstruction for certain quadratic fibrations where each fiber is compact