NONEXTENSIVE STATISTICAL MECHANICS AND ITS APPLICATIONS
TABLE OF CONTENTS I NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS HISTORICAL BACKGROUND AND PRESENT STATUS 1 II QUANTUM DENSITY MATRIX DESCRIPTION OF NONEXTENSIVE SYSTEMS 97 III TSALLIS THEORY THE MAXIMUM ENTROPY PRINCIPLE AND EVOLUTION EQUATIONS 155 IV COMPUTATIONAL METHODS FOR THE SIMULATION OF CLASSICAL AND QUANTUM MANY BODY SYSTEMS ARISING FROM NONEXTENSIVE THERMOSTATISTICS 190 V CORRELATION INDUCED BY NONEXTENSIVITY AND THE ZEROTH LAW OF THERMODYNAMICS 231 VI DYNAMIC AND THERMODYNAMIC STABILITY OF NONEXTENSIVE SYSTEMS 237 VII GENERALIZED SIMULATED ANNEALING ALGORITHMS USING TSALLIS STATISTICS APPLICATION TO J SPIN GLASS MODEL 247 VIII PROTEIN FOLDING SIMULATIONS BY A GENERALIZED ENSEMBLE ALGORITHM BASED ON TSALLIS STATISTICS 253 I NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS HISTORICAL BACKGROUND AND PRESENT STATUS ABSTRACT THE DOMAIN OF VALIDITY OF STANDARD THERMODYNAMICS AND BOLTZMANN GIBBS STATISTICAL MECHANICS IS FOCUSED ON ALONG A HISTORICAL PERSPECTIVE IT IS THEN FORMALLY ENLARGED IN ORDER TO HOPEFULLY COVER A VARIETY OF ANOMALOUS SYSTEMS THE GENERALIZATION CONCERNS NONEXTENSIVE SYSTEMS WHERE NONEXTENSIVITY IS UNDERSTOOD IN THE THERMODY NAMICAL SENSE THIS GENERALIZATION WAS RST PROPOSED IN 1988 INSPIRED BY THE PROBA BILISTIC DESCRIPTION OF MULTIFRACTAL GEOMETRY AND HAS BEEN INTENSIVELY STUDIED DURING THIS DECADE IN THE PRESENT E ORT WE DESCRIBE THE FORMALISM DISCUSS THE MAIN IDEAS AND THEN EXHIBIT THE PRESENT STATUS IN WHAT CONCERNS THEORETICAL EXPERIMENTAL AND COMPUTATIONAL EVIDENCES AND CONNECTIONS AS WELL AS SOME PERSPECTIVES FOR THE FUTURE THE WHOLE REVIEW CAN BE CONSIDERED AS AN ATTEMPT TO CLARIFY OUR CURRENT UNDERSTANDING OF THE FOUNDATIONS OF STATISTICAL MECHANICS AND ITS THERMODYNAMICAL IMPLICATIONS 1 INTRODUCTION THE PRESENT E ORT IS AN ATTEMPT TO REVIEW IN A SELF CONTAINED MANNER A ONE DECADE OLD NONEXTENSIVE GENERALIZATION 1 2 OF STANDARD STATISTICAL MECHANICS AND THERMODYNAMICS AS WELL AS TO UPDATE AND DISCUSS THE RECENT ASSOCIATED DE VELOPMENTS 3 CONCOMITANTLY WE SHALL ADDRESS ON PHYSICAL GROUNDS THE DOMAIN OF VALIDITY OF THE BOLTZMANN GIBBS BG FORMALISM I E UNDER WHAT RESTRICTIONS IT IS EXPECTED TO BE VALID ALTHOUGH ONLY THE DEGREE OF UNIVERSALITY OF BG THERMAL STATISTICS WILL BE FOCUSED ON LET US RST MAKE SOME GENERIC COMMENTS IN SOME SENSE EVERY PHYSICAL PHENOMENON OCCURS SOMEWHERE AT SOME TIME 4 CONSISTENTLY THE ULTIMATE MOST PROBABLY UNATTAINABLE GOAL OF PHYSICAL SCIENCES IS IN WHAT THEORY CONCERNS TO DEVELOP FORMALISMS THAT APPROACH AS MUCH AS POSSIBLE UNIVERSALITY I E VALID FOR ALL PHENOMENA UBIQUITY I E VALID EVERYWHERE AND ETERNITY I E VALID ALWAYS SINCE THESE WORDS ARE VERY RICH IN MEANINGS LET US ILLUSTRATE WHAT WE SPECI CALLY REFER TO THROUGH THE BEST KNOWN PHYSICAL FORMALISM NAMELY NEWTONIAN MECHANICS AFTER ITS PLETHORIC VERI CA 1 UNIVERSAL INDEED WE ALL KNOW THAT WHEN THE INVOLVED VELOCITIES APPROACH THAT OF LIGHT IN THE VACUUM NEWTONIAN MECHANICS BECOMES ONLY AN APPROXIMATION AN INCREASINGLY BAD ONE FOR VELOCITIES INCREASINGLY CLOSER TO THAT OF LIGHT AND REALITY APPEARS TO BE BETTER DESCRIBED BY SPECIAL RELATIVITY ANALOGOUSLY WHEN THE INVOLVED MASSES ARE AS SMALL AS SAY THE ELECTRON MASS ONCE AGAIN NEWTO NIAN MECHANICS BECOMES BUT A BAD APPROXIMATION AND QUANTUM MECHANICS BECOMES NECESSARY TO UNDERSTAND NATURE ALSO IF THE INVOLVED MASSES ARE VERY LARGE NEWTONIAN MECHANICS HAS TO BE EXTENDED INTO GENERAL RELATIVITY TO SAY IT IN OTHER WORDS WE KNOW NOWADAYS THAT WHENEVER 1 C INVERSE SPEED OF LIGHT IN VACUUM AND OR H PLANCK CONSTANT AND OR G GRAVITATIONAL CONSTANT ARE DI ERENT FROM ZERO NEWTONIAN MECHANICS IS STRICTLY SPEAKING FALSE SINCE IT ONLY CONSERVES AN ASYMPTOTIC VALIDITY ALONG THESE LINES WHAT CAN WE SAY ABOUT BG STATISTICAL MECHANICS AND STAN DARD THERMODYNAMICS A DI USE BELIEF EXISTS AMONG NOT FEW PHYSICISTS AS WELL AS OTHER SCIENTISTS THAT THESE TWO INTERCONNECTED FORMALISMS ARE ETERNAL UBIQUI TOUS AND UNIVERSAL IT IS CLEAR THAT AFTER MORE THAN ONE CENTURY HIGHLY SUCCESSFUL APPLICATIONS OF STANDARD THERMODYNAMICS AND THE MAGNI CENT BOLTZMANN S CON NECTION OF CLAUSIUS MACROSCOPIC ENTROPY TO THE THEORY OF PROBABILITIES APPLIED TO THE MICROSCOPIC WORLD BG THERMAL STATISTICS CAN AND SHOULD EASILY BE CON SIDERED AS ONE OF THE PILLARS OF MODERN SCIENCE CONSISTENTLY IT IS CERTAINLY FAIR TO SAY THAT BG THERMOSTATISTICS AND ITS ASSOCIATED THERMODYNAMICS ARE ETERNAL AND UBIQUITOUS IN PRECISELY THE SAME SENSE THAT WE HAVE USED ABOVE FOR NEW TONIAN MECHANICS BUT AGAIN IN COMPLETE ANALOGY WITH NEWTONIAN MECHANICS WE CAN BY NO MEANS CONSIDER THEM AS UNIVERSAL IT IS UNAVOIDABLE TO THINK THAT LIKE ALL OTHER CONSTRUCTS OF HUMAN MIND THESE FORMALISMS MUST HAVE PHYSICAL RESTRICTIONS I E DOMAINS OF APPLICABILITY OUT OF WHICH THEY CAN AT BEST BE BUT APPROXIMATIONS THE PRECISE MATHEMATICAL DE NITION OF THE DOMAIN OF VALIDITY OF THE BG STA TISTICAL MECHANICS IS AN EXTREMELY SUBTLE AND YET UNSOLVED PROBLEM FOR EXAMPLE THE ASSOCIATED CANONICAL EQUILIBRIUM DISTRIBUTION IS CONSIDERED A DOGMA BY TAK ENS 5 SUCH A RIGOROUS MATHEMATICAL APPROACH IS OUT OF THE SCOPE OF THE PRESENT E ORT HERE WE SHALL FOCUS ON THIS PROBLEM IN THREE STEPS THE RST ONE IS DEEPLY RELATED TO KRYLOV S PIONEERING INSIGHTS 6 SEE ALSO 7 9 INDEED KRYLOV ARGUED HALF A CENTURY AGO THAT THE PROPERTY WHICH STANDS AS THE HARD FOUNDATION OF BG STATISTICAL MECHANICS IS NOT ERGODICITY BUT MIXING MORE PRECISELY QUICK ENOUGH EXPONENTIAL MIXING I E POSITIVE LARGEST LIAPUNOV EXPONENT WE SHALL REFER TO SUCH SITUATION AS STRONG CHAOS THIS CONDITION WOULD ESSENTIALLY GUARAN TEE PHYSICALLY SHORT RELAXATION TIMES AND WE BELIEVE THERMODYNAMIC EXTENSIVITY WE ARGUE HERE THAT WHENEVER THE LARGEST LIAPUNOV EXPONENT VANISHES WE CAN HAVE SLOW TYPICALLY POWER LAW MIXING SEE ALSO 8 9 SUCH SITUATIONS WILL BE REFERRED AS WEAK CHAOS IT IS EXPECTED TO BE ASSOCIATED WITH ALGEBRAIC INSTEAD OF EXPONENTIAL RELAXATIONS AND TO THERMODYNAMIC NONEXTENSIVITY HOPEFULLY FOR LARGE CLASSES OF ANOMALOUS SYSTEMS OF THE TYPE DESCRIBED IN THE PRESENT REVIEW THE SECOND STEP CONCERNS THE QUESTION OF WHAT GEOMETRICAL STRUCTURE CAN BE RESPONSIBLE FOR THE MIXING BEING OF THE EXPONENTIAL OR OF THE ALGEBRAIC TYPE THE 2 PICTURE WHICH EMERGES DETAILS WILL BE SEEN LATER ON IS THAT WHEN THE PHYSICALLY RELEVANT PHASE SPACE OR ANALOGOUS QUANTUM CONCEPT IS SMOOTH EUCLIDEAN LIKE IN THE SENSE THAT IT IS CONTINUOUS DI ERENTIABLE ETC THE MIXING IS OF THE EXPONENTIAL TYPE IN CONTRAST IF THAT SPACE HAS A MULTIFRACTAL STRUCTURE THEN THE MIXING BECOMES KIND OF UNEASY AND TENDS TO BE OF THE ALGEBRAIC OR EVEN SLOWER TYPE THE THIRD AND LAST STEP CONCERNS THE QUESTION OF WHAT KIND OF PHYSICAL CIR CUMSTANCES CAN PRODUCE A SMOOTH TRANSLATIONALLY INVARIANT IN SOME SENSE OR INSTEAD A MULTIFRACTAL SCALING INVARIANT IN SOME SENSE STRUCTURE FOR THE RELE VANT PHASE SPACE AT RST APPROXIMATION THE SCENARIO SEEMS TO BE AS FOLLOWS IF THE E ECTIVE MICROSCOPIC INTERACTIONS ARE SHORT RANGED I E CLOSE SPATIAL CON NECTIONS AND THE E ECTIVE MICROSCOPIC MEMORY IS SHORT RANGED I E CLOSE TIME CONNECTIONS FOR INSTANCE MARKOVIAN LIKE PROCESSES AND THE BOUNDARY CONDITIONS ARE SMOOTH NON MULTI FRACTAL AND THE INITIAL CONDITIONS ARE STANDARD ONES AND NO PECULIAR MESOSCOPIC DISSIPATION OCCURS E G LIKE THAT OCCURRING IN VARIOUS TYPES OF GRANULAR MATTER ETC THEN THE ABOVE MENTIONED SPACE IS SMOOTH AND BG STA TISTICAL MECHANICS APPEARS TO CORRECTLY DESCRIBE NATURE IF ONE OR MORE OF THESE RESTRICTIONS IS VIOLATED LONG RANGE INTERACTIONS AND OR IRREDUCIBLY NONMARKO VIAN MICROSCOPIC MEMORY 10 AND OR MULTIFRACTAL BOUNDARY CONDITIONS AND OR QUITE PATHOLOGICAL INITIAL CONDITIONS ARE IMPOSED AND OR SOME SPECIAL TYPES OF DISSIPATION ARE ACTING ETC THEN THE ABOVE MENTIONED SPACE CAN BE MULTIFRAC TALLY STRUCTURED AND ANOMALOUS NONEXTENSIVE STATISTICAL MECHANICS SEEM TO BE NECESSARY TO DESCRIBE NATURE SEE ALSO 11 TO SUMMARIZE THE OVERALL PICTURE WE MAY SAY ROUGHLY SPEAKING THAT A SMOOTH RELEVANT PHASE SPACE TENDS TO CORRESPOND TO BG STATISTICAL MECHANICS EXPONENTIAL MIXING ENERGY DEPENDENCE OF THE CANONICAL EQUILIBRIUM DISTRIBUTION I E THE CELEBRATED BOLTZMANN FACTOR AND TIME DEPENDENCE OF TYPICAL RELAXATION PROCESSES AND EXTENSIVE THERMODYNAMICS ENTROPY THERMODYNAMIC POTENTIALS AND SIMILAR QUANTITIES PROPORTIONAL TO THE NUMBER OF MICROSCOPIC ELEMENTS OF THE SYSTEM IN CONTRAST A MULTIFRACTALLY STRUCTURED PHASE SPACE TENDS TO COR RESPOND TO ANOMALOUS STATISTICAL MECHANICS HOPEFULLY FOR AT LEAST SOME OF THE TYPICAL SITUATIONS HEREIN DESCRIBED IN SOME DETAIL POWER LAW MIXING ENERGY DEPENDENCE OF THE CANONICAL EQUILIBRIUM DISTRIBUTION AND TIME DEPENDENCE OF TYPICAL RELAXATION PROCESSES AND NONEXTENSIVE THERMODYNAMICS ANOMALOUS EN TROPY THERMODYNAMIC POTENTIALS AND SIMILAR QUANTITIES THE BASIC GROUP OF SYMMETRIES WOULD BE CONTINUOUS TRANSLATIONS OR ROTATIONS IN THE RST CASE AND DILATATIONS IN THE SECOND ONE THIS OPENS OF COURSE THE DOOR TO EVEN MORE GENERAL SCENARIOS RESPECTIVELY ASSOCIATED TO MORE COMPLEX GROUPS OF SYMMETRIES 12 BUT AGAIN THIS IS OUT OF OUR PRESENT SCOPE THE ACTUAL SITUATION IS NATURALLY EXPECTED TO BE MORE COMPLEX AND CROSS IMBRICATED THAT THE ONE JUST SKETCHED HERE BUT THESE WOULD NEVERTHELESS BE THE ESSENTIAL GUIDING LINES BEFORE ENTERING INTO THE NONEXTENSIVE THERMOSTATISTICAL FORMALISM HEREIN AD DRESSED LET US MENTION AT LEAST SOME OF THE THERMODYNAMICAL ANOMALIES THAT WE HAVE IN MIND AS PHYSICAL MOTIVATIONS AS ARGUED ABOVE IT IS NOWADAYS QUITE WELL KNOWN THAT A VARIETY OF PHYSICAL SYSTEMS EXISTS FOR WHICH THE POWERFUL AND 3