Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View

In this paper, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics: 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 questions, we first introduce measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. And we propose the measurement theoretical foundation of equilibrium statistical mechanics, and further, answer the above 1) and 2), that is, 1) is “No”, but, 2) is “Yes”.


Introduction
Recently in [1][2][3][4][5][6] we proposed (classical and quantum) measurement theory, which is characterized as the linguistic (or, metaphysical) turn of quantum mechanics.As seen in [1][2][3][4][5][6], this theory includes several conventional system theories (e.g, quantum system theory, statistics, dynamical system theory and so on).Also, for the philosophical aspect of measurement theory (called the quantum mechanical world view), see [5].And thus, we believe that measurement theory is one of the most fundamental theories in science.
Note that there are several opinions ( cf. [2,3,7-9] ) 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 probability and the ergodic hypothesis in the light of measurement theory [4,5] (i.e., Axioms 1 and 2, Interpretation (E) mentioned in the following section).

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 mechanics (or, as a linguistic turn of quantum mechanics), constructed as the mathematical theory formulated in a certain -algebra * C A (i.e., a norm closed subalgebra in the operator algebra   B H composed of all bounded operators on a Hilbert space H, cf.[10,11] ) as follows: Axiom 2 [MT] = causality measurement  For completeness, note that measurement theory (A) is not physics but a kind of language based on "the quantum mechanical world view" (cf.[5]).When = ( ) A B H , the -algebra composed of all compact operators on a Hilbert space H, the (A) is called quantum measurement theory (or, quantum system theory), 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 complex-valued functions vanishing at infinity on a locally compact Hausdorff space [10])), the (A) is called classical measurement theory.Thus, we have the following classification: Hence, we consider that (B 2 ) the theory of classical mechanical world view = classical measurement theory in (B 1 )  MT (i.e., the theory of quantum mechanical world view).And we never consider that (B 3 ) the theory of classical mechanical world view = 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 (B 3 ).Thus, our interest in this paper may be regarded as our method [2,3] in (B 2 ) versus Ruelle's method [7] can be also identified with (called a spectrum space or maximal ideal space) such as In this sense, the is also called a state space in classical measurement theory.


Here, assume that the -algebra A B H  has the identity I .This assumption is not unnatural, since, if I A  , it suffices to reconstruct the above A such that it includes   A I  .According to the noted idea (cf.[13]) in quantum mechanics, an observable , where 0 and I is the 0-element and the identity in A respectively.3): for any 1 2 , F    such that , it holds that .
For the further argument (e.g.,   -field, countably additivity, the -algebraic formulation, etc.), see [4,6].* W With any system , a -algebra * 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 A  S and an observable is represented by an observable O := ( , , ) X F F in A. Also, the measurement of the observable for the system with the state ).An observer can obtain a measured value x ( X  ) by the measurement The Axiom 1 presented below is a kind of mathematical generalization of Born's probabilistic interpretation of quantum mechanics.And thus, it is a statement without reality.
. easurement The probability that a measured value x ( X  ) obtained by the measurement Next, we explain Axiom 2 in (A).Let be a tree, i.e., a partial ordered set such that 1 3 t and 2 3 ( , is called a Markov relation ( due to the Heisenberg picture), if it satisfies the following conditions (D 1 ) and (D 2 ).
, a Markov operator 1 2 2 , : The family of dual operators : the Markov relation is said to be deterministic.Now Axiom 2 in the measurement theory (A) is presented as follows: Further, we have to explain how to use Axioms 1 and 2 as follows.That is, we present the following interpretation (E) [=(E 1 ) -(E 4 )], which is characterized as a kind of linguistic turn of so-called Copenhagen interpretation.That is, we propose ( cf. [4,5]): (E 1 ) Consider the dualism composed of observer and system( =measuring object).And therefore, observer and system must be absolutely separated.
(E 2 ) 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.
(E 3 ) Also, the observer does not have the space-time.Thus, the question: When and where is a measured value obtained? is out of measurement theory.
Thus, we say that (E 4 ) 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 (E 3 ) says, for example, that Schrödinger's cat is out of the world of measurement theory.

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 problem: (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.

About 1)
In Newtonian mechanics, any state of a system composed of particles is represented by a point (position, momentum) = 24 ( 10 ) in a phase (or state) space n n n q q q p p , , , , , = ., , 2 particle mass Fix .And define the measure > 0 where    be the flow on the energy surface E  induced by the Newton equation with the Hamiltonian H, or equivalently, Hamilton's equation: Liouville's theorem ( cf. [9]) says that the measure E  is invariant concerning the flow .Defining the normalized measure we have the normalized measure space   , ,    , we define the deterministic Markov relation
For example, consider one particle P 1 .Put   a state  such that the particle 1 always stays a corner of the box}.Clearly, it holds that , then the particle 1 must always stay a corner.This contradicts 2).Therefore, 2) means the following: The ergodic theorem (cf.[14]) says that the above 2)' is equivalent to the following equality: After all, the ergodic property says that if T is su- Copyright © 2012 SciRes.WJM fficiently large, it holds that , , , ) is called a (normalized) first staying time space, also, the probability space   , ,  is called a (normalized) second staying time space.Note that these mathematical probability spaces are not related to probability" ( cf.Section 3.2).

Put
. For each , define the coordinate map such that q q q p p p q q q p p p    for all : where   # K the number of the elements of the set K.
And, we regard   for any The following important remark was missed in [2,3].This is the advantage of our method in comparison h Ruelle's method ( cf. [7]).
. Hence, in order to expect that 3)' and 4)' hold, it suffices to consider that 5 T  seconds.Also, we see, by (7) and (5), that, for Particularly, putting 0 0, Hence, we can describe the 3) and 4) in terms of in what follows.
   be as in the above.Then, summing up 3) and 4), by (9) we have: random va ble wi al distribution in the pproximately independen ria s th the identic sense that there exists Also, a state   3.


for almost all t.That is, ).Then, from Hypothesis A, th cf.[12]) says t e law of large numbers ( hat e t.Consid for almost all tim er the decomposition ), where 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 hypothesis" such that in the abstra uivalently thesis should d that the er ), ( ) 1 L t p t   ould be "population average of N particles at each t" = "time average of one particle" Thus, we can assert that the ergodic hypothesis is related 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 entropy
= log : , , , , , , , q q q p p p   of the particle belongs In what follows, we shall represent this (J) in terms of measurements.Define the observable . Then we say, by Axiom 1, that sured value obtained (K) the probability that the mea by the measurement  wever, Int at it suffices to take the simultaneous measurement Remark 4. [The principle of equal a priori probabilities].The (J) (or equivalently, (K)) says choose a particle from , and not choose a state from the state space E  equal .Thus, as mentioned in the abstract, the principle of (a priori) probability is not related to with the approximately identical distribution concerning time.In other words, there exists a normalized measure E  on (i.e., ) such that:

wit.
Remark 2. [About the time interval   0,T ].For example, as one of typical cases, consider the motion of 10 24 particles in a cubic box (whose long side is 0.3 m).It is usual to consider that averaging velocity = And therefore, the collisions rarely happen among   0 # K particles in the time interval   0,T , and therefore, the motion is "almost independent".For example, putting   every time t).