World Journal of Mechanics, 2012, 2, 125130 doi:10.4236/wjm.2012.22014 Published Online April 2012 (http://www.SciRP.org/journal/wjm) Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View Shiro Ishikawa Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan Email: ishikawa@math.keio.ac.jp Received December 20, 2011; revised January 25, 2012; accepted February 28, 2012 ABSTRACT In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical me chanics: 1): Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics? 2): Is the ergodic hypothesis related to equilibrium statistical mechanics? Note that these problems are not yet answered, since there are several opinions for the formulation of equilibrium statistical mechanics. In order to answer the above ques tions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is character ized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equili brium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”. Keywords: The Copenhagen Interpretation; Probability; Operator Algebra; Ergodic Theorem; Quantum and Classical Measurement Theory; Liouville’s Theorem; The Law of Increasing Entropy 1. Introduction Recently in [16] we proposed (classical and quantum) measurement theory, which is characterized as the lin guistic (or, metaphysical) turn of quantum mechanics. As seen in [16], this theory includes several conventional system theories (e.g, quantum system theory, statistics, dynamical system theory and so on). Also, for the phi losophical aspect of measurement theory (called the qu antum mechanical world view), see [5]. And thus, we be lieve that measurement theory is one of the most fun damental theories in science. Note that there are several opinions ( cf. [2,3,79] ) for the formulation of equilibrium statistical mechanics, and hence, there are several opinions for the problems 1) and 2) mentioned in the abstract. The purpose of this paper is to reinforce our method [2, 3], or equivalently, to clarify the principle of equal pro bability and the ergodic hypothesis in the light of mea surement theory [4,5] (i.e., Axioms 1 and 2, Interpreta tion (E) mentioned in the following section). 2. Measurement Theory (Axioms 1 and 2, Interpretation) In this section, according to [4], we explain the outline of measurement theory (or in short, MT). Measurement theory is, by an analogy of quantum me chanics (or, as a linguistic turn of quantum mechanics), constructed as the mathematical theory formulated in a certain algebra * C (i.e., a norm closed subalgebra in the operator algebra B composed of all bounded operators on a Hilbert space H, cf. [10,11] ) as follows: (A) (language) Axiom 1Axiom 2 [MT] =causality measurement For completeness, note that measurement theory (A) is not physics but a kind of language based on “the quan tum mechanical world view” (cf. [5]). When =() c BH, the algebra composed of all compact operators on a Hilbert space H, the (A) is called quantum measurement theory (or, quantum system the ory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when A is commutative (that is, when A is characterized by , the algebra composed of all continuous complexvalued functions vanishing at infinity on a locally compact Hausdorff space * C 0 C* C (cf. [10])), the (A) is called classical measure ment theory. Thus, we have the following classification: (B1) 0 quantum MT(when) MT classical MT(when) C AB AC Hence, we consider that (B2) the theory of classical mechanical world view = classical measurement theory in (B1) MT (i.e., the theory of quantum mechanical world view). And we never consider that (B3) the theory of classical mechanical world view Copyright © 2012 SciRes. WJM
S. ISHIKAWA 126 = Something like Newtonian mechanics + Kolmogorov’s probability theory [12], which may be usually called dynamical system theory. It should be noted that Ruelle’s method (cf. [7]), which is the most authorized approach to equilibrium statistical mechanics, is based on the (B3). Thus, our interest in this paper may be regarded as our method [2,3] in (B2) versus Ruelle’s method [7] in (B3). Now we shall explain measurement theory (A). Let AB be a algebra, and let * C* be the dual Banach space of A. That is, A* ={ is a continuous linear functional on }, and the norm * A is defined by () = sup:suchthat1 BH AF F F FA . Define the mixed state * such that *=1 A and 0 forallhat0F Fsuch tFA . And put * *=. is amixed state mA A S For example, note that 0 * 1 mm M C S = { is a measure on such that =1 }. A mixed state * m S is called a pure state if it satisfies that 12 =1 * for some 12 ,m S and 0< <1 implies 12 == . Put * p * = m , is apurestate AA SS which is called a state space. It is well known (cf. [10]) that *= pc BH S{uu (i.e., the Dirac notation) =1 p S H u, and * 0 C= 0 0 is a point measure at 0 , where 0 0 f d f 0 fC . The latter implies that C S* p 0 can be also identified with (called a spectrum space or maximal ideal space) such as * 0(spectrum space) (state space) pC S (1) In this sense, the is also called a state space in classical measurement theory. Here, assume that the algebra * C AB has the identity . This assumption is not unnatural, since, if A, it suffices to reconstruct the above A such that it includes A . According to the noted idea (cf. [13]) in quantum mechanics, an observable O:= ,, FF in is defined as follows: (C1) [Field] is a set, , the power set of (2 X F ) is a field of , that is, “12 ,1 2 F ”, “\ XF ”. (C2) [Finite additivity] is a mapping from F to A satisfying: 1): for every , is a nonnegative element in F such that 0 I , 2): =0F and = I X, where 0 and I is the 0element and the identity in A respectively. 3): for any 12 , such that , it holds that . 12 = FF F 121 2 For the further argument (e.g., field, countably additivity, the algebraic formulation, etc.), see [4, 6]. * W With any system , a algebra S* C AB * can be associated in which the measurement theory (A) of that system can be formulated. A state of the system S is represented by an element p S and an ob servable is represented by an observable O:=( , , ) FF in A. Also, the measurement of the observable for the system with the state O S is denoted by [] O, MAS (or more precisely, [] MAXF O:= ,S ,,F). An observer can obtain a mea sured value x ( ) by the measurement [] O, MAS . The Axiom 1 presented below is a kind of mathema tical generalization of Born’s probabilistic interpretation of quantum mechanics. And thus, it is a statement with out reality. 1Axiom . easurement The probability that a mea sured value x ( ) obtained by the measurement [0] O:= ,S XF M,, F A belongs to a set is given by 0F . Next, we explain Axiom 2 in (A). Let be a tree, i.e., a partial ordered set such that 13 t and 23 (,T t )tt implies 1 tt 2 or 21 tt . Assume that there exists an element 0 tT , called the root of T, such that 0 tt (tT ) holds. Put 1 2 t t 2=T2 12 ,Ttt . The fa mily AA 2 12 , 121 (,) ttt tt T 2 t : is called a Markov re lation ( due to the Heisenberg picture), if it satisfies the following conditions (D1) and (D2). (D1) With each tT , a algebra * Ct is associ ated. (D2) For every 2 12 ,T tt , a Markov operator 12 2 ,: tt t1 t A = 13 ,tt is defined. And it satisfies that 12 23 ,,tt tt holds for any , 12 ,tt 2 23 ,tt T . The family of dual operators 12 2 ** ,2 (, ) 12 :mm tt t tt T AA 1 * t SS is called a dual Mar kov relation (due to the Schrödinger picture). Also, when 12 12 ** , p tt tt SS p A* A holds for any 2 12 ,T tt , the Markov relation is said to be deterministic. Now Axiom 2 in the measurement theory (A) is pre sented as follows: 2 Causali .Axiom ty The causality is represented by a Markov relation 1221 12 ,2 (,) : tttttt T AA . Further, we have to explain how to use Axioms 1 and 2 as follows. That is, we present the following interpre tation (E) [=(E1) – (E4)], which is characterized as a kind of linguistic turn of socalled Copenhagen interpretation. That is, we propose ( cf. [4,5]): (E1) Consider the dualism composed of observer and system( =measuring object). And therefore, observer and Copyright © 2012 SciRes. WJM
S. ISHIKAWA 127 system must be absolutely separated. (E2) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it can not be measured any longer. Also, the causality should be assumed only in the side of system, however, a state never moves. Thus, the Heisenberg picture should be adopted. (E3) Also, the observer does not have the spacetime. Thus, the question: When and where is a measured value obtained? is out of measurement theory. Thus, we say that (E4) there is no probability without measurement. Since measurement theory is a kind of language, the spirit is based on Wittgenstein’s famous statement: “the limits of my language mean the limits of my world”. Thus, the (E3) says, for example, that Schrödinger’s cat is out of the world of measurement theory. 3. Equilibrium Statistical Mechanics in Measurement Theory 3.1. Statements Concerning Axiom 2 (Dynamical Aspect; Ergodic Hypothesis) 3.1.1. Equilibrium Statistical Mechanical Phenomena Assume that about particles (for example, hydrogen molecules) move in a box. It is natural to assume the following phenomenas 1)4) 24 (10)N 1) Every particle obeys Newtonian mechanics. 2) Every particle moves uniformly in the box. For example, a particle does not halt in a corner. 3) Every particle moves with the same statistical behavior concerning time. 4) The motions of particles are (approximately) independent of each other. In what follows we shall devote ourselves to the pro blem: (F) how to describe the above equilibrium statistical mechanical phenomenas 1)4) in terms of measurement theory. For completeness, again note that measurement theory is a kind of language. 3.1.2. Abo u t 1 ) In Newtonian mechanics, any state of a system composed of particles is represented by a point (position, momentum) = 24 (10)N(,)qp 3=1 , ) (123 12 (,, ,, nn ) in a phase (or state) space nn n qqq pp 6nn p 6 :N H. Let be a Hamiltonian such that 123 123 =1 2 123 =1 =1 =1,2,3 ,,,, , =. ,, 2particle mass N nnnn nn n NN kn nnn n nk Hqqq ppp pUqqq (2) Fix . And define the measure >0E on the energy surface 6= ,, N E E qp qp such that 1 61 =d , , theBorelfieldof E EN B Hm qp B BB (3) where 1/2 22 =1 =1,2,3 = , N nk kn kn HH Hqp pq and 61N dm is the usual surface measure on . Let << E tt be the flow on the energy surface in duced by the Newton equation with the Hamiltonian H, or equivalently, Hamilton’s equation: dd =, = dd . =1, 2,3,=1, 2,, kn kn kn kn qp , H tptq knN (4) Liouville’s theorem ( cf. [9]) says that the measure is invariant concerning the flow . Defi ning the normalized measure << E tt such that E EEE , we have the normalized measure space ,, E E B . Putting 0 == E AC C (from the compact ness of ), , , =T =, t qp tt 122 1 ,=E ttt t , 12 , 12 11 * ,( tt tt tt ) = 1 tE , we define the deter ministic Markov relation 12 12 ,2 (,) : ttEEtt T CC in Axiom 2. 3.1.3. Abo u t 2 ) Now let us begin with the wellknown ergodic theorem ( cf. [9, 14]). For example, consider one particle P1. Put 1 P= S a state such that the particle 1 always stays a corner of the box}. Clearly, it holds that 1 P E S . Also, if ES 11 tP , then the particle 1 must always stay a corner. This contra dicts 2). Therefore, 2) means the following: P S 0<t P 2)’ [Ergodic property]: If a compact set ,= E SS satisfies E tS S 0<t , then it holds that = S . The ergodic theorem (cf. [14]) says that the above 2)’ is equivalent to the following equality: 0 ((state) space average)(time average) 0 1 = dd lim ,(), E TE Et T EE f t T fC After all, the ergodic property says that if T is su Copyright © 2012 SciRes. WJM
S. ISHIKAWA 128 fficiently large, it holds that 0 1. d d E TE Et f t T (5) Put d = d Tt mtT. The probability space [, ] ,, ,TT Bm T (or equivalently, [0, ] [0, ],, TT TB m ) is called a (normalized) first staying time space, also, the probability space ,, E E B is called a (normalized) second staying time space. Note that these mathematical probability spaces are not related to probability” ( cf. Section 3.2). ) Remark 1. [About 2)’]. In [2,3], we started form the mathematical statement 2)’. In this paper, this is im proved by the phenomenological statement 2). 3.1.4. Ab out 3) and 4) Put . For each 24 =1, 2,,10 N KN ( kK 6 N , de fine the coordinate map such that 6 kE π: 123 123 =1 123 123 π()=π(, ) =π,,,, , =,,,,, kk N knnnn nn n kkkk kk qp qqq ppp qqq pp p (6) for all 123 123 =1 6 =,,,,, , nnnn nn n N E qqq ppp qp Also, for any subset K , define the distribution map 24 =1,2, ,10 N KN () 6 1 :m N KE D 6 M such that (, )6 (, ) 1 =(,) # qp N KqpE k kK Dqp K where # is the number of the elements of the set K. Let 0 be a state. For each nK, we de fine the map such that 06 :0, nT X 0 0 =π0, E nn t Xt T t (7) And, we regard 0 =1 nn X as random functions on the probability space [0, ] ,, 0, TT Bm T . Then, 3) and 4) respectively means 3)’ 0 =1 nn X is a sequence with the approximately identical distribution concerning time. In other words, there exists a normalized measure on (i.e., ) such that: 6 6 1 m EM 0 6 : 0, ,= 1, 2,, T n mtX Tt Bn N 0 =1 nn X for any E (8) 4)’ is approximately independent, in the sense that, 24 01,2,,(10) KN such that 0 1# N (that is, 0 #0 K N), it holds that . : 0, B tX T 06 06 0 0 :, 0, Tkk Tkk kK B mtX kK Tt mt The following important remark was missed in [2,3]. This is the advantage of our method in comparisonh Ruelle’s method ( cf. [7]). wit Remark 2. [About the time interval 0,T]. For exam ple, as one of typical cases, consider the motion of 1024 particles in a cubic box (whose long side is 0.3 m). It is usual to consider that averaging velocity = 2 510ms, mean free path = 7 10 m . And therefore, the collisions rarely happen among 0 # particles in the time inter val 0,T, and therefore, the motion is “almost inde pendent”. For example, putting 10 0 #=10K, we can calculate the number mes a certain particle collides with of ti 0 particles in [0,T] as 1 24 2 7 10 10 510 510 10 10 TT 5 . Hence, in order to expect that 3)’ and 4)’ hold, it suffices to consider that 5T seconds. Also, we see, by (7) and (5), that, for 0N K such that 0 1# N, 6 0 0 0 1 00 1 0 1 0 :, :π(, 0, π =: 0, π . π E kt k Ek k TtkK kK k k EkK kK k k EkK kK BkK BkK T mtT (9) Particularly, putting 0 0, Tkk mtX Tt 60 00 =T mt 0=Kk, we see: 01 :π 0, TkEk mtX Tt 6. B (10) Hence, we can describe the 3) and 4) in terms of =1 π nn in what follows. Hypothesis A [3) and 4)]. Put Le 24 1, 2,,(10) KN . N t H, E, , , 6 π: kE be as in the above. Then, summing up 3) and 4), by (9) we have: (G) 6 =1 π: kEk is at random vablewial distribution in the pproximately independen rias th the identic Copyright © 2012 SciRes. WJM
S. ISHIKAWA 129 sense that there exists 6 1 m EM such that 0 1 π kK E (11) 0 = p . E kkK “ for all roduct measure” and 0 1# N 0 K. Also, a state , qp (, ) N qp is called an state if it satisfies equilibrium E D . 3.1.5. Eypothergodic Hsis Ne following th ic hypothesis]. Assume Hypothesis ). Then, for any ow, we have theorem ( cf. [2,3]): Theorem A [Ergod A (or equivalently, 3) and 4) =(0),(0) 0 qp , it holds that 0 ((),()) 24 : 0, ,= 1, 2,(10) N qt pt KTk DmtX Tt Bk N (12) 6 for almost all t. That is, , 0 :(12) 0, T mtdoes not hold T 1. Proof. Let 0 K such that 00 1# NN (that is, 0 0 0 # # 1 N ). Then, from Hypothesis A, th cf. [12]) says te law of large numbers (hat 0 ((),t()) 1 π q pt KEk D (13) e t. Considfor almost all timer the decomposition KN = (1)(2)( ) ,,, L KK K. (i.e., () =1 =L l l K , ()()== ll KK ll =1,2, ,lL. From ), where (13), () #l KN it holds that, 0 for each k 24 =1,2,,(10 )N, () =1 1 () , l l l K (( ) ((),()) () =1 # = 1#π q qt pt K Kl N L EEkE l KD DN N (14) for almost all time t. Thus, by (10), we get (12). Hence, the proof is completed. We believe that Theorem A is just what sh represented by the “ergodic h ypothesis” such that in the abstrauiva lently thesis shouldd that the er ), () 1Lt pt ould be “population average of N particles at each t” = “time average of one particle” Thus, we can assert that the ergodic hypothesis is re lated to equilibrium statistical mechanics (cf. the 2) ct). Here, the ergodic property 2)’ (or eq rgodic hypo, equality (5)) and the above e not be confused. Also, it should be note godic hypothesis does not hold if the box (containing particles) is too large. Remark 3 [The law of increasing entropy]. The en tropy , Hqp of a state , qp is defined by (, )(,) =log : ,, NN qpq p EK KE Hk DD qpq p where 3 =! tant annconstant N kN Plank cons Boltzm Since almost every state in is equilibrium, the entropy of almost every state is equal log E k . that the law of in Therefore, it is natural to assum creasing entropy holds. 3. ent) echanics. For completeness, note t related to pr ment. cal system at almost all time t can be re ga he particles whose states belong to e 2. Statements Concerning Axiom 1 (Probabilistic Aspect; Measurem In this section we shall study the probabilistic aspects of equilibrium statistical m that (H) the argument in the previous section is no obability since Axiom 1 does not appear in Section 3.1. Also, recall the (E4), that is, there is no probability without measure Note that the (12) implies that the equilibrium stati stical mechani rded as: (I) a box including about 1024 particles such as the number of t 6 B is given by 24 10 E . Thus, it is natural to assume as follows. (J) if we , at random, choose a particle from 1024 par ticles in the box at time t, then the probability that the state 6 123 123 ,,,, ,qqq pp p of the particle belongs 6 is given b By to E . In what follows, we shall represent this (J) in terms of measurements. Define the observable 6 6 0 0,, O= BF in C such that 0 6 , # N K N qp K B . N E 6 # ,, 15 k qp Thus, we have the measurement (, )π, =qp kqp D F 6 00 6 0 0 () (,) ,, O:= , ME Cqp t BFS . Then we say, by Axiom 1, that sured value obtained (K) the probability that the mea by the measurement 6 00 6 0 0 () (,) ,, O:= , ME Cqp t BFS belongs to 6 B is given by E . That is because Theo rem A says that 000 , t Fqp E (almost every time t). Copyright © 2012 SciRes. WJM
S. ISHIKAWA Copyright © 2012 SciRes. WJM 130 Also, let : E t E CC erator determined by ion 3.1 , we must take a for each tim Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277306. doi:10.1016/S01650114(96)001145 beic Markov opthe coap e (E2) says th e, Examples 1 and 3 in [4]. N particles in box a determinist ntinuous m : E tE E (cf. Sect.2). Then, it clearly holds 00 O=O E t . And 0 [( () O, ME CS [2] S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., Tokyo, 2006. ( ),( ))] kk qt pt Ho erpretation 12 ,,,, , kn tt tt. [3] S. Ishikawa, “Ergodic Problem in Quantitative Language,” Far East Journal of Dynamical Systems, Vol. 11, No. 1, 2009, pp. 3348. wever, Intat it suffices to take the simultaneous measurement [ ] ( (0),(0)) ,qp S . Here, for the simultaneous O n k , see, for instanc 0 =1 () O ME n k C observable 0 =1 [4] S. Ishikawa, “A New Interpretation of Quantum Mecha nics,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 3542. [5] S. Ishikawa, “Quantum Machanics and the Philosophy of Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, No. 1, 2012, pp. 29. Remark 4. [The principle of equal a priori probabi lities]. The (J) (or equivalently, (K)) says choose a par ticle from, and not choose a state from the state space equal . Thus, as mentioned in the abstract, the principle of (a priori) probability is not related to [6] S. Ishikawa, “A Measurement Theoretical Foundation of Statistics,” Journal of Applied Mathematics, Vol. 3, No. 3, 2012, pp. 283292. our method. If we try to describe Ruele’s method [7] in terms of measurement theory, we must use statistical measurement theory ( cf. [2,6]). However, this trial will end in failure. Also, our recent report [15] will promote the understanding of measurement theory. 4. Conclusions Our concern in this paper may be regarded as the pro blem: “What is the classical mechanical world view?” Concretely speaking, we are concerned wit [7] D. Ruelle, “Statistical Mechanics, Rigorous Results,” World Scientific, Singapore, 1969. [8] G. Gallavotti, “Statistical Mechanics: A Short Treatise,” Springer Verlag, Berlin, 1999. [9] M. Toda, R. Kubo and N. Saito, “Statistical physics, Springer Series in SolidState Sciences,” Springer Verlag, Berlin, 1983. [10] G. J. Murphy, “C*Algebras and Operator Theory,” Aca demic Press, Waltham, 1990. h the problem: hus, “our method [2,3] vs. Ru aper, we added important remarks “(B2) vs. (B3)”, and t method [7]”. In this p ele’s [11] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932. (i.e., Remarks 1 and 2) to our method [2,3], and streng thened our method in the light of the mechanical world view [4,5]. Equilibrium statistical mechanics is of course one of the most fundamental theories in science. And it is sure that Ruele’s method [7] has been authorized for a long time. Therefore, we hope that our proposal will be exa mined from various view points. [12] A. N. Kolmogorov, “Foundations of the Theory of Prob ability (Translation),” 2nd Edition, Chelsea Pub Co, New York, 1960. [13] E. B. Davies, “Quantum Theory of Open Systems,” Aca demic Press, Waltham, 1976. [14] U. Krengel, “Ergodic Theorems,” Walter de Gruyter, Berlin, 1985. doi:10.1515/9783110844641 [15] S. Ishikawa, “The Linguistic Interpretation of Quantum Mechanics,” 2012. http://arxiv.org/abs/1204.3892 REFERENCES [1] S. Ishikawa, “A Quantum Mechanical Approach to a
