With the help of a general expression of the entropies in extensive and nonextensive systems, some important relations between thermodynamics and statistical mechanics are revealed through the views of thermodynamics and statistic physics. These relations are proved through the MaxEnt approach once again. It is found that for a reversible isothermal process, the information contained in the first and second laws of thermodynamics and the MaxEnt approach is equivalent. Moreover, these relations are used to derive the probability distribution functions in nonextensive and extensive statistics and calculate the generalized forces of some interesting systems. The results obtained are of universal significance.
We study the thermodynamic properties of solid and metal electrons in the nonextensive quantum statistics with a nonextensive parameter transformation. First we study the nonextensive grand canonical distribution function and the nonextensive quantum statistics with a parameter transformation. Then we derive the generalized Boson distribution and Fermi distribution in the nonextensive quantum statistics. Further we study the thermodynamic properties of solid and metal electrons in the nonextensive quantum system, including the generalized Debye models, the generalized internal energies, the ge…
We show that the zeroth law of thermodynamics holds within an alternative version of nonextensive statistical mechanics based on {\it incomplete probability distribution}. The generalized zeroth law leads to a generalized definition of thermodynamic functions which are possible to be used for systems with important nonextensivity (nonadditivity) in energy, volume or other external variables.
Within Tsallis' nonextensive statistics, a model is elaborated to address self-similar time series as a thermodynamic system. Thermodynamic-type characteristics relevant to temperature, pressure, entropy, internal and free energies are introduced and tested. Stability conditions of time series analysis are discussed in details on the basis of Van der Waals model.
We have constructed a nonextensive thermodynamic formalism consisting of two sets of parallel Legendre transformation structures in previous papers. One is the physical set and the other is the Lagrange set. In this paper we study the thermodynamic stability criterion with a dual interpretation of the thermodynamic relations and quantities. By recourse to the assumption that volume in nonextenstive system is nonadditive, we conclude that it is the physical pressure that is responsible for the mechanical balance between any two parts in a given nonextensive system. It is verified that in the ph…
Macroscopic nonextensive thermodynamics is studied without recourse to microscopic statistical mechanics. It is shown that if entropy is nonextensive, the concept of physical temperature introduced through the generalized zeroth law of thermodynamics necessarily leads to modifications of the first law of thermodynamics and some of thermodynamic relations including Clausius definition of thermodynamic entropy. It is also shown, by applying this generalized Clausius entropy to a composite nonextensive system, how the nonextensive entropy and the quantity of heat consistently behave in an arbitra…
The concepts of quantity of heat and work are deduced in the non-extensive statistical mechanics context, following steps in parallel to those employed in the extensive statistical mechanics.
Tsallis' thermostatistics is by now recognized as a new paradigm for statistical mechanical considerations. However, it is still affected by a serious hindrance: the generalization of thermodynamics' zero-th law to a nonextensive scenario is plagued by difficulties. Here we show how to overcome this problem.
In this paper, Landau diamagnetism is revisited in the context of nonextensive statistical thermodynamics. Exact as well as approximate expressions for partition function, magnetization and susceptibility are obtained by using Hilhorst integral transformation which provides exact solutions in the framework of Tsallis statistics. The results have been reassessed with the standard results to illustrate the effect of nonextensivity.
The nonextensive statistical ensembles are revisited for the complex systems with long-range interactions and long-range correlations. An approximation, the value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted for the limit of large particle number for most normal systems. In this case, Tsallis entropy can be expanded as a function of energy and particle number fluctuation, and thus the power-law forms of the generalized Gibbs distribution and grand canonical distribution can be derived. These new distribution functions can be applied to derive the free energy and grand…
We show an attempt to establish the zeroth law of thermodynamics within the framework of nonextensive statistical mechanics based on the classic normalization $\texttt{Tr}\hatρ=1$ and the unnormalized expectation $x=\texttt{Tr}\hatρ^q\hat{x}$. The first law of thermodynamics and the definition of heat and work in this formalism are discussed.
Within the nonextensive framework, it is shown that zeroth law of thermodynamics determines not only the mapping between Lagrange multipliers and intensive variables, but also the mapping between nonextensive and extensive entropy. The form of constraints decides the form of the extensive entropy, standard averages lead to Boltzmann-Shannon-Gibbs entropy while normalised biased averages lead to Renyi entropy. The mapping between Lagrange multipliers and intensive variables is also discussed in the more general context of composable entropy.
Based on the Tsallis entropy, the nonextensive thermodynamic properties are studied as a q-deformation of classical statistical results using only probabilistic methods and straightforward calculations. It is shown that the constant in the Tsallis entropy depends on the deformation parameter and must be redefined to recover the usual thermodynamic relations. The notions of variance and covariance are generalised. A partial derivative formula of the entropy is established. It verifies important relations from which most of the nonextensive thermodynamics relations can be recovered. This leads t…
An instructive and apparently simple model of fully-coupled rotators, the so-called Hamiltonian Mean Field (HMF) model, together with a generalized version with variable interaction range, have revealed a very complex out-of-equilibrium dynamics which can be considered paradigmatic for nonextensive systems. In this short paper we illustrate the interesting anomalous pre-equilibrium dynamics, focusing on the novel connections to the generalized nonextensive thermostatistics and the recent links to glassy systems.
The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range of q between zero and two. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.
Recently, Gross claims that Boltzmann entropy $S=k\ln W$ is valid for any system at equilibrium, so that Tsallis entropy is useless in this case. I comment on some arguments forwarded to reach this conclusion and argue that the additive energy formalism of nonextensive statistics is not appropriate for the fundamental study of the theory for nonadditive systems.
The nonextensive thermodynamic relations are expressed under the assumption of temperature duality, endowing the "physical temperature" and the "Lagrange temperature" in different physical sense. Based on this assumption, two sets of parallel Legendre transform structures are given. One is called "physical" set, and the other called "Lagrange" set. In these two formalisms, the thermodynamic quantities and the thermodynamic relations are both liked through the Tsallis factor. Application of the two sets of the thermodynamic relations to the self-gravitating system shows that the heat capacity d…
The generalized zeroth law of thermodynamics indicates that the physical temperature in nonextensive statistical mechanics is different from the inverse of the Lagrange multiplier, beta. This leads to modifications of some of thermodynamic relations for nonextensive systems. Here, taking the first law of thermodynamics and the Legendre transform structure as the basic premises, it is found that Clausius definition of the thermodynamic entropy has to be appropriately modified, and accordingly the thermodynamic relations proposed by Tsallis, Mendes and Plastino [Physica A 261 (1998) 534] are als…
Within the Tsallis thermodynamics' framework, and using scaling properties of the entropy, we derive a generalization of the Gibbs-Duhem equation. The analysis suggests a transformation of variables that allows standard thermodynamics to be recovered. Moreover, we also generalize Einstein's formula for the probability of a fluctuation to occur by means of the maximum statistical entropy method. The use of the proposed transformation of variables also shows that fluctuations within Tsallis statistics can be mapped to those of standard statistical mechanics.
We comment on letter ``Is Tsallis Thermodynamics Nonextensive?'' by E. Vives and A. Planes [Phys. Rev. Lett. 88, 020601 (2002) cond-mat/0106428]. It is pointed out that the Euler and Gibbs-Duhem equations derived in the letter can serve to justify an appropriate form for the Lagrange parameters controlling thermal equilibrium, without need of any change of variables. This leads to a framework for Tsallis Thermodynamics which is free from recent criticisms raised by Nauenberg [Phys. Rev. E 67, 036114 (2003) cond-mat/0210561] and Gross [Physica (Amsterdam) 305, 99 (2002) cond-mat/0106496]. This …
We review some of the most recent results on the dynamics of the Hamiltonian Mean Field (HMF) model, a systems of N planar spins with ferromagnetic infinite-range interactions. We show, in particular, how some of the dynamical anomalies of the model can be interpreted and characterized in terms of the weak-ergodicity breaking proposed in frameworks of glassy systems. We also discuss the connections with the nonextensive thermodynamics proposed by Tsallis.
We study the nonextensive thermodynamics for open systems. On the basis of the maximum entropy principle, the dual power-law q-distribution functions are re-deduced by using the dual particle number definitions and assuming that the chemical potential is constant in the two sets of parallel formalisms, where the fundamental thermodynamic equations with dual interpretations of thermodynamic quantities are derived for the open systems. By introducing parallel structures of Legendre transformations, other thermodynamic equations with dual interpretations of quantities are also deduced in the open…
We investigate perturbative thermodynamic geometry of nonextensive ideal Classical, Bose and Fermi gases.We show that the intrinsic statistical interaction of nonextensive Bose (Fermi) gas is attractive (repulsive) similar to the extensive case but the value of thermodynamic curvature is changed by nonextensive parameter. In contrary to the extensive ideal classical gas, the nonextensive one may be divided to two different regimes. According to deviation parameter of the system to the nonextensive case, one can find a special value of fugacity, $z^{*}$, where the sign of thermodynamic curvatur…
The present work investigates the Lesche stability (experimental robustness), the thermodynamic stability, the Legendre structure of thermodynamics, and derives the Maximum Entropy distribution of the one--parametric ``nonextensive Gaussian'' entropy. We show that this entropy definition fulfills both stability conditions for all values of its parameter ($q\in\mathbb{R}$). The entropy maximizer contains the Lambert $W$--function, which allows the preservation of the Legendre transformations.
The generalized Gibbs free energy and enthalpy is derived in the framework of nonextensive thermodynamics by using the so-called physical temperature and the physical pressure. Some thermodynamical relations are studied by considering the difference between the physical temperature and the inverse of Lagrange multiplier. The thermodynamical relation between the heat capacities at a constant volume and at a constant pressure is obtained using the generalized thermodynamical potential, which is found to be different from the traditional one in Gibbs thermodynamics. But, the expressions for the h…
Starting from the basic prescriptions of the Tsallis' nonextensive thermostatistics, i.e. generalized entropy and normalized q-expectation values, we study the relativistic nonextensive thermodynamics and derive a Boltzmann transport equation that implies the validity of the H-theorem where a local nonextensive four-entropy density is considered. Macroscopic thermodynamic functions and the equation of state for a perfect gas are derived at the equilibrium.
The problem of factorization of a nonextensive probability distribution is discussed. It is shown that, in general, the correlation energy between the correlated subsystems in the canonical composite system can not be neglected even in the thermodynamic limit. In consequence, the factorization approximation should be employed carefully according to different systems. It is also shown that the zeroth law of thermodynamics can be established in the framework of the Incomplete Statistical Mechanics (ISM).
The forms of Euler and Gibbs-Duhem relations discussed in thermodynamics of extensive systems are shown to hold also for nonextensive systems with long-range interactions with a novel interpretation of entities appearing therein. In this way, the principles underlying Tsallis' scaling relations in equilibrium nonextensive thermostatistics are clarified.
We consider the behavior of electrons in an external uniform magnetic field B where the space coordinates perpendicular to B are taken as noncommuting. This results in a generalization of standard thermodynamics. Calculating the susceptibility, we find that the usual Landau diamagnetism is modified. We also compute the susceptibility according to the nonextensive statistics of Tsallis for (1-q)<<1, in terms of the factorization approach. Two methods agree under certain conditions.
The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns {\it nonextensive} systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly describe the formalism, and then exhi…
The $q$-deformed statistics for fermions arising within the non-extensive thermostatistical formalism has been applied to the study of various quantum many-body systems recently. The aim of the present note is to point out some subtle difficulties presented by this approach in connection with the problem of thermodynamic consistency. Different possible ways to apply the $q$-deformed quantum distributions in a thermodynamically consistent way are considered.
The incomplete nonextensive statistics in the canonical and microcanonical ensembles is explored in the general case and in a particular case for the ideal gas. By exact analytical results for the ideal gas it is shown that taking the thermodynamic limit, with $z=q/(1-q)$ being an extensive variable of state, the incomplete nonextensive statistics satisfies the requirements of equilibrium thermodynamics. The thermodynamical potential of the statistical ensemble is a homogeneous function of the first degree of the extensive variables of state. In this case, the incomplete nonextensive statistic…
In this lecture we briefly review the definition, consequences and applications of an entropy, $S_q$, which generalizes the usual Boltzmann-Gibbs entropy $S_{BG}$ ($S_1=S_{BG}$), basis of the usual statistical mechanics, well known to be applicable whenever ergodicity is satisfied at the microscopic dynamical level. Such entropy $S_q$ is based on the notion of $q$-exponential and presents properties not shared by other available alternative generalizations of $S_{BG}$. The thermodynamics proposed in this way is generically {\it nonextensive} in a sense that will be qualified. The present frame…
We show that the zeroth principle of thermodynamics applies to aging quasistationary states of long-range interacting $N$-body Hamiltonian systems. We also discuss the measurability of the temperature in these out-of-equilibrium states using a {\it short-range} interacting thermometer. As many connections are already established between such quasistationary states and nonextensive statistical mechanics, our results are the first evidence that such basic concepts apply to systems that the nonextensive formalism aims to describe.
In this paper, we show that 1) additive energy is not appropriate for discussing the validity of Tsallis or Rényi statistics for nonextensive systems at meta-equilibrium; 2) $N$-body systems with nonadditive energy or entropy should be described by generalized statistics whose nature is prescribed by the existence of thermodynamic stationarity. 3) the equivalence of Tsallis and Rényi entropies is in general not true.
A nonextensive thermostatic approach to chaotic dynamical systems is developed by expressing generalized Tsallis distribution as escort distribution. We explicitly show the thermodynamic limit and also derive the Legendre Transform structure. As an application, bit variance is calculated for ergodic logistic map. Consistency of the formalism demands a relation between box size ($ε$) and degree of nonextensivity, given as $(1-q)\sim -1/{\rm ln} ε$. This relation is numerically verified for the case of bit variance as well as using basic definition of Tsallis entropy.
This is a study of the information evolution of complex systems by geometrical consideration. We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness of the state number counting at any scale on fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper, where $q$ is the fractal dimension divided by the dimension of the smooth Euclidean space in which the fractal structure of the phase space is embedded. I…
We present a stability analysis of the classical ideal gas in a new theory of nonextensive statistics and use the theory to understand the phenomena of negative specific heat in some self-gravitating systems. The stability analysis is made on the basis of the second variation of Tsallis entropy. It is shown that the system is thermodynamically unstable if the nonextensive parameter is q>5/3, which is exactly equivalent to the condition of appearance of the negative specific heat.
In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for $1<q<\frac{3}{2}$ and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.
It is argued that the factorization of compound probability over subsystems is a consequence of the existence of thermodynamic equilibrium in the composite system having Tsallis entropy. So it should be respected by all exact calculations concerning interacting nonextensive subsystems. Using nonextensive energy satisfying the factorization, we propose an additive formalism of nonextensive statistical mechanics with additive q-deformed physical quantities and exponential distributions. This formalism leads to exact quantum gas distributions different from those given by factorization approximat…
Recently, in Phys. Rev. Lett. 95, 140601 (2005), P. Grassberger addresses the interesting issue of the applicability of q-statistics to the renowned Feigenbaum attractor. He concludes there is no genuine connection between the dynamics at the critical attractor and the generalized statistics and argues against its usefulness and correctness. Yet, several points are not in line with our current knowledge, nor are his interpretations. We refer here only to the dynamics on the attractor to point out that a correct reading of recent developments invalidates his basic claim.
The general formalism for the nonextensive statistics based on the Landsberg-Vedral entropy was derived. The formula for the first law of thermodynamics and the exact relations of the thermodynamic quantities to their ensemble averages were obtained. It was found that under the transformation $q\to 2-q$ the probabilities of microstates of the nonextensive statistics based on the Landsberg-Vedral entropy formally resemble the corresponding probabilities of the Tsallis statistics with escort probabilities. However, the nonextensive statistics with the Landsberg-Vedral entropy does not require in…
We study the dynamics of a Hamiltonian system of N classical spins with infinite-range interaction. We present numerical results which confirm the existence of metaequilibrium Quasi Stationary States (QSS), characterized by non-Gaussian velocity distributions, anomalous diffusion, Lévy walks and dynamical correlation in phase-space. We show that the Thermodynamic Limit (TL) and the Infinite-Time Limit (ITL) do not commute. Moreover, if the TL is taken before the ITL the system does not relax to the Boltzmann-Gibbs equilibrium, but remains in this new equilibrium state where nonextensive thermo…
We obtain the specific heat in the third constraint scenario for a canonical ensemble of a nonextensive extreme relativistic ideal gas in a closed form. The canonical ensemble of N particles in D dimensions is well-defined for the choice of the deformation parameter in the range 0 < q < 1 + 1 / DN. For a relativistic ideal gas with particles of arbitrary mass a perturbative scheme in the nonextensivity parameter (1 - q) is developed by employing an infinite product expansion of the q-exponential, and a direct transformation of the internal energy from the second to the third constraint picture…
Following the basic prescriptions of the Tsallis' nonextensive thermodynamics, we study the relativistic nonextensive thermodynamics and the equation of state for a perfect gas at the equilibrium. The obtained results are used to study the relativistic nuclear equation of state in the hadronic and in the quark-gluon plasma phase. We show that small deviations from the standard extensive statistics imply remarkable effects into the shape of the equation of state.
We have shown that the weak-coupling limit superconductors are well described by $ q \sim 1 $, where $ q $ is a real parameter which characterizes the degree of nonextensivity of Tsallis' entropy. Nevertheless, small deviations with respect to q=1 provide better agreement when compared with experimental results. We have also shown that the generalized BCS theory with $ q \neq 1 $ exhibit power-law behavior of several measurable macroscopic functions in the low-temperature regime. These power-law properties are found in many high-Tc oxides superconductors and motivated us to extend Tsallis' ent…
We extend the formalism of the thermodynamic two-time Green's functions to nonextensive quantum statistical mechanics. Working in the optimal Lagrangian multipliers representation, the $q$-spectral properties and the methods for a direct calculation of the two-time $q$% -Green's functions and the related $q$-spectral density ($q$ measures the nonextensivity degree) for two generic operators are presented in strict analogy with the extensive ($q=1$) counterpart. Some emphasis is devoted to the nonextensive version of the less known spectral density method whose effectiveness in exploring equili…
We derive and study quasicanonical Gibbs distribution function which is characterized by the thermostat with finite number of particles (quasithermostat). We show that this naturally leads to Tsallis nonextensive statistics and thermodynamics, with Tsallis parameter q is found to be related to the number of particles in the quasithermostat. We show that the chi-square distribution of fluctuating temperature used recently by Beck can be partially understood in terms of normal random momenta of particles in the quasithermostat. Also, we discuss on the importance of the time scale hierarchy and f…
Thermodynamics of power means applies to an ideal quantum gas which may be nonextensive. Transition to an ideal classical gas occurs when the empirical temperature exponents of the internal energy and absolute temperature coalesce. Limiting processes are pure heat conduction and pure deformations. Largest and smallest mean final volumes occur for isothermal and adiabatic processes, respectively. The increment in the heat admits two integrating factors which yield conserved quantities for adiabatic processes. Energy-conserving equilibrations yield the largest final means possible, while the sec…
Schrödinger suggested that thermodynamical functions cannot be based on the gratuitous allegation that quantum-mechanical levels (typically the orthogonal eigenstates of the Hamiltonian operator) are the only allowed states for a quantum system [E. Schrödinger, Statistical Thermodynamics (Courier Dover, Mineola, 1967)]. Different authors have interpreted this statement by introducing density distributions on the space of quantum pure states with weights obtained as functions of the expectation value of the Hamiltonian of the system. In this work we focus on one of the best known of these distr…
In this note it is shown that in his ``Comments'', Tsallis did not point out any flaws in the main criticism of my paper, namely, that the nonextensive q-entropy formalism fails to satisfy the zeroth law of thermodynamics. Details are also given of a rigorous thermodynamic proof which demonstrates that, contrary to Tsallis's assertion, the application of a nonextensive formalism to black-body radiation does not lead to the well known $T^4$ Stefan-Boltzman law.
Canonical ensembles consisting of $M$-unit $Hubbard$ $dimers$ have been studies within the nonextensive statistics (NES). The temperature dependences of the energy, entropy, specific heat and susceptibility have been calculated for the number of dimers, $M = 1, 2, 3$ and $\infty$. We have assumed the relation between the entropic index $q$ and the cluster size $N$ given by $q=1+2/N$ ($N = 2\:M$ for $M$ dimers), which was previously derived by several methods. For relating the physical temperature $T$ to the Lagrange multiplier $β$, two methods have been adopted: $T=1/k_B β$ in the method A [Ts…
In the case of a system with an unbounded hamiltonian the entropic index q of non-extensive thermodynamics has an upperbound q_c>1 beyond which the formalism becomes meaningless. The expression 1/(q_c-1) is the dimension of the state space (i.e. the manifold of density matrices) in the context of non-commutative geometry. For q=q_c an ultraviolet cutoff E< E_ub is needed to guarantee the existence of the equilibrium density matrix. Duality between q>1 and q<1-statistics via a q --> 1/q transformation is established. It leads to an overall picture in which the meaning of both q>1-statistics and…
The nonextensive statistics based on Tsallis entropy have been so far used for the systems composed of subsystems having same $q$. The applicability of this statistics to the systems with different $q$'s is still a matter of investigation. The actual difficulty is that the class of systems to which the theory has been applied is limited by the usual nonadditivity rule of Tsallis entropy which, in reality, has been established for the systems having same $q$ value. In this paper, we propose a more general nonadditivity rule for Tsallis entropy. This rule, as the usual one for same $q$-systems, …
In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonext…
The thermodynamical property of a small cluster including $M$ Hubbard dimers, each of which is described by the two-site Hubbard model, has been discussed within the nonextensive statistics (NES). We have calculated the temperature dependence of the energy, entropy, specific heat and susceptibility for $M = 1, 2, 3$ and $\infty$, assuming the relation between the entropic index $q$ and the cluster size $N$ given by $q=1+2/N$ ($N = 2 M$ for $M$ dimers), which was previously derived by several methods. For relating the physical temperature $T$ to the Lagrange multiplier $β$, two methods have bee…
The nonextensive statistics based on the $q$-entropy $S_q=-\frac{\sum_{i=1}^v(p_i-p_i^q)}{1-q}$ has been so far applied to systems in which the $q$ value is uniformly distributed. For the systems containing different $q$'s, the applicability of the theory is still a matter of investigation. The difficulty is that the class of systems to which the theory can be applied is actually limited by the usual nonadditivity rule of entropy which is no more valid when the systems contain non uniform distribution of $q$ values. In this paper, within the framework of the so called incomplete information th…
Thermodynamical properties of canonical and grand-canonical ensembles of the half-filled two-site Hubbard model have been discussed within the framework of the nonextensive statistics (NES). For relating the physical temperature $T$ to the Lagrange multiplier $β$, two methods have been adopted: $T=1/k_B β$ in the method A [Tsallis {\it et al.} Physica A {\bf 261} (1998) 534], and $T=c_q/k_B β$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281} (2001) 126], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, $p_i$ the probability distribution of the $i$th state, and $q$ the e…
