{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T15:36:31Z","timestamp":1775057791478,"version":"3.50.1"},"reference-count":90,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2012,1,30]],"date-time":"2012-01-30T00:00:00Z","timestamp":1327881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Rate-Controlled Constrained-Equilibrium (RCCE) method for the description of the time-dependent behavior of dynamical systems in non-equilibrium states is a general, effective, physically based method for model order reduction that was originally developed in the framework of thermodynamics and chemical kinetics. A generalized mathematical formulation is presented here that allows including nonlinear constraints in non-local equilibrium systems characterized by the existence of a non-increasing Lyapunov functional under the system\u2019s internal dynamics. The generalized formulation of RCCE enables to clarify the essentials of the method and the built-in general feature of thermodynamic consistency in the chemical kinetics context. In this paper, we work out the details of the method in a generalized mathematical-physics framework, but for definiteness we detail its well-known implementation in the traditional chemical kinetics framework. We detail proofs and spell out explicit functional dependences so as to bring out and clarify each underlying assumption of the method. In the standard context of chemical kinetics of ideal gas mixtures, we discuss the relations between the validity of the detailed balance condition off-equilibrium and the thermodynamic consistency of the method. We also discuss two examples of RCCE gas-phase combustion calculations to emphasize the constraint-dependent performance of the RCCE method.<\/jats:p>","DOI":"10.3390\/e14020092","type":"journal-article","created":{"date-parts":[[2012,1,30]],"date-time":"2012-01-30T11:10:39Z","timestamp":1327921839000},"page":"92-130","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics"],"prefix":"10.3390","volume":"14","author":[{"given":"Gian Paolo","family":"Beretta","sequence":"first","affiliation":[{"name":"Department of Mechanical and Industrial Engineering, Universita di Brescia, Brescia, 25123, Italy"}]},{"given":"James C.","family":"Keck","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}]},{"given":"Mohammad","family":"Janbozorgi","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Northeastern University Boston, MA 02115, USA"}]},{"given":"Hameed","family":"Metghalchi","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Northeastern University Boston, MA 02115, USA"}]}],"member":"1968","published-online":{"date-parts":[[2012,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1063\/1.1700223","article-title":"The induction period in chain reactions","volume":"20","author":"Benson","year":"1952","journal-title":"J. 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The Maximum Entropy Principle, MIT Press."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1086\/260506","article-title":"Expectations and exchange rate dynamics","volume":"84","author":"Dornbusch","year":"1976","journal-title":"J. Polit. Econ."},{"key":"ref_62","unstructured":"Differently from [17] where a DKM is required to show whether the SIM is orthogonal to the constant entropy contours, for the RCCE method this depends on the features of the entropy and constraining functions. We can show this by considering the simplest case of a two dimensional x-y state space, with F = S(x,y) and a single constraining functional C\u03c4(x,y) = y, so that the constraint is y = c\u03c4. To maximize S(x,y) + \u03b3\u03c4(y \u2212 c\u03c4) we need Sx = 0 and Sx = \u2212\u03b3\u03c4 (subscripts here denote partial derivatives). So, on the x-y plane the RCCE trajectory is rce(c\u03c4) = xce(c\u03c4)i + c\u03c4j where xce(c\u03c4) is defined by Sx(xce,c\u03c4) = 0. By differentiating Sx(xce,c\u03c4) = 0, we also get \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t                  \u02d9\n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                ce\n\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t       = (\u2212Sxy\/Sxx) \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                  c\n\t\t\t\t\t\t\t\t\t\t\t                  \u02d9\n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                \u03c4\n\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t       and therefore the vector \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                  r\n\t\t\t\t\t\t\t\t\t\t\t                  \u02d9\n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                ce\n\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t       = [(\u2212Sxy\/Sxx)i + j]\n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                  c\n\t\t\t\t\t\t\t\t\t\t\t                  \u02d9\n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                \u03c4\n\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t      . Since the entropy gradient vector is \u2207S = Sxi + Syj = \u2212\u03b3\u03c4j, we note that \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                  r\n\t\t\t\t\t\t\t\t\t\t\t                  \u02d9\n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                ce\n\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t       is in general not in the direction of the entropy gradient and that we reach this conclusion independently of the DKM which determines the rate \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t      \n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                  c\n\t\t\t\t\t\t\t\t\t\t\t                  \u02d9\n\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t                \u03c4\n\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t      . This geometrical feature of RCCE shows that the method lives at a different level than the reduction schemes which can draw conclusions only from the underlying DKM."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"160","DOI":"10.3390\/entropy-e10030160","article-title":"Modeling non-equilibrium dynamics of a discrete probability distribution: General rate equation for maximal entropy generation in a maximum-entropy landscape with time-dependent constraints","volume":"10","author":"Beretta","year":"2008","journal-title":"Entropy"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/S0034-4877(09)90024-6","article-title":"Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes","volume":"64","author":"Beretta","year":"2009","journal-title":"Rep. Math. Phys."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"012004","DOI":"10.1088\/1742-6596\/237\/1\/012004","article-title":"Maximum entropy production rate in quantum thermodynamics","volume":"237","author":"Beretta","year":"2010","journal-title":"J. Phys. Conf."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Dewar, R., Lineweaver, C., Niven, R., and Regenauer-Lieb, K. (2012). Beyond The Second Law: Entropy Production and Non-Equilibrium Systems, Springer-Verlag. Available online: http:\/\/www.quantumthermodynamics.org\/.","DOI":"10.1007\/978-3-642-40154-1_1"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1007\/BF00949059","article-title":"Chemical reactions and the principle of maximal rate of entropy production","volume":"34","author":"Ziegler","year":"1983","journal-title":"J. Appl. Math. Phys. (ZAMP)"},{"key":"ref_68","unstructured":"For example, these could be the populations or occupation probabilities of the different electronic levels in an atom, or they could be the set of densities (energy density, number density for each type of species) describing the local equilibrium state of fluid elements in a continuous reacting mixture."},{"key":"ref_69","unstructured":"The Second Law ([36], p. 62) also requires that if the equilibrium states with the same value of the energy are more than one, then one of these must be stable equilibrium whereas all the others must be nonstable equilibrium."},{"key":"ref_70","unstructured":"To fix ideas, in addition to the examples we develop explicitly in Section 4, it is useful to consider the following important examples in contexts other than chemical equilibrium and chemical kinetics. In statistical mechanics, the (dimensionless) Boltzmann-Gibbs entropy is such a state functional, \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t      F\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  S\n\t\t\t\t\t\t\t\t\t\t\t\t                  \u02dc\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \n\t\t\t\t\t\t\t\t\t\t\t\t                \u2211\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t                  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  1\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  n\n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  ln\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t      , where xj are probabilities associated with the energy levels of the system. In quantum statistical mechanics (see, e.g., [84] and references therein) and quantum thermodynamics (see, e.g., [85] and references therein), the (dimensionless) von Neumann entropy functional is \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t      F\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  S\n\t\t\t\t\t\t\t\t\t\t\t\t                  \u02dc\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \n\t\t\t\t\t\t\t\t\t\t\t\t                \u2211\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t                  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  1\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  n\n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  ln\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  Tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u03c1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  ln\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u03c1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t      , where xj are the eigenvalues (repeated if degenerate) of the density or quantum-state operator \u03c1, and Tr denotes the trace functional. In information theory [86], the Shannon uncertainty is such a state functional, \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t      F\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t                H\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \n\t\t\t\t\t\t\t\t\t\t\t\t                \u2211\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t                  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  1\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  n\n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  log\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t      , where xj are probabilities associated with a set of possible outcomes. In control theory, the total error variance is such a functional, \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t      F\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t                W\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \n\t\t\t\t\t\t\t\t\t\t\t\t                \u2211\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t                  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  1\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  n\n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t      , where xj are deviations from target values such as for example a desired movement trajectory. In non-extensive statistical mechanics [87,88] (see also, e.g., [89,90]), which captures important statistical features in spectroscopy, turbulence, and financial markets, the Havrda-Charvat-Tsallis entropy is such a functional, \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t      F\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  S\n\t\t\t\t\t\t\t\t\t\t\t\t                  \u02dc\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                (\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  )\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  1\n\t\t\t\t\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t\t\t\t\t                    1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  q\n\t\t\t\t\t\t\t\t\t\t\t\t                  \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  [\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \u2212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \n\t\t\t\t\t\t\t\t\t\t\t\t                \u2211\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t                  =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  1\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  n\n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t              \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t                  x\n\t\t\t\t\t\t\t\t\t\t\t\t                  j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  q\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  ]\n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t      , where xj are probabilities associated with a set of possible outcomes and q is the index of the corresponding L\u00e9vy distribution (or van der Waals spectroscopic broadening). This is a generalization of the usual entropy functional, in the sense that \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  S\n\t\t\t\t\t\t\t\t\t\t\t\t                  \u02dc\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                \u2192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t                  S\n\t\t\t\t\t\t\t\t\t\t\t\t                  \u02dc\n\t\t\t\t\t\t\t\t\t\t\t\t                \n\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t        \n\t\t\t\t\t\t\t\t\t\t\t\t       in the limit q \u2192 1. 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