Ergodic Hypothesis in Physics and Democracy: a Comparative Analysis J

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ISSN 2277-2685 IJESR/Feb. 2018/ Vol-8/Issue-2/6-9 J. Saikia et. al., / International Journal of Engineering & Science Research

ERGODIC HYPOTHESIS IN PHYSICS AND DEMOCRACY: A COMPARATIVE ANALYSIS J. Saikia*1, R. Mahanta2, M. Konwar3, G.D. Baruah4 1Department of Physics J.B College, Jorhat, Assam, India. 2Deaprtment of Physics, Dibru College, Dibrugarh, Assam, India. 3Principal, Moran College, Assam, India. 4Centre for laser and Optical Science, New Uchamati, Doom Dooma, Assam, India. ABSTRACT In the present work we discuss the topic of ergodic hypothesis in physics in connection with its relation with a non physics topic like democracy in political science. In many other ways the ergodic hypothesis in physics is analogous to the democratic system in polities. The work may be useful for understanding Ergodic hypothesis from analogies.

Keywords: Canonical distribution, phase space, ergodicity. 1. INTRODUCTION Ergodic hypothesis which is the basic of statistical mechanics (a branch of physics) is a hypothesis concerning phase space and has been introduced by L. Boltzmann in 1870. In this connection it is worthwhile to mention the name of Josiah Willard Gibbs (1839-1903) who developed the theory of chemical thermodynamics and was one of the founders of statistical mechanics. Although in general it is very different to prove ergodic or quasi ergodic hypothesis for a given system, it has been reliably found to be correct by repeated trails. Al through ergodic hypothesis is considered by some people as irrelevant it has been discussed extensively in physics. Consideration of ergodicity occurs in dynamics since the behavior can be complex even for simple dynamical systems. Now it is appropriate to introduce the word democracy which is the constant source of discussion among politicians. Democracy is a system of government in which the people elect their leader or country with this system. We find it appropriate here to remark that in many ways some phenomena in physics appearing in different contexts are quite analysis in some cases to phenomena that show up in none physics contexts. In the present work we have reported a comparative study between the system of democracy and ergodic hypothesis. Survey of available literatures indicate that this the first report of its kind. The root of the analogy is the fact that both the ergodic hypothesis (in physics) and the concept of democracy are to a conceptual idealization, probabilistic theories. Both involve huge numbers. In the following sections we proceed to define few words or terms like phase space, ergodic hypothesis so as to make our analysis in the form of comparative study with the phenomenon (?) of democracy is made more amenable. Our belief is that all physical phenomena must follow specific laws of nature. This is also applicable to non physics phenomena like democracy. 2. PHASE SPACE The words phase space is repeatedly used in physics to describe a particular system consisting of a large number of particles. For a system with-n degree of freedom, the 2n-dimendional space with coordinates (q1, q2, q3 …… qn ; p1, p2, p3 ,….. pn), where the q’s describe the degree of freedom and p’s are the corresponding momenta. Each point represent a state of the system. In a gas of N-point particles each particle has there positional coordinates and three corresponding momenta coordinators, so that the phase space has 6N-dimensions. If the particles have internal degree of freedom, such as the vibrations and rotation molecules, then these must be included in the phase space, which is consequently of higher dimension than that of point particles. As the system changes with time the representative point trace out a curve in phase space known as a trajectory. In this connection it is worthy of referring to the attractors is statistical physics, which are the set of points in “phase space” to which the representative point of a dissipative system tends as the system evolves. The attraction can be a single point, *Corresponding Author www.ijesr.org 6

J. Saikia et. al., / International Journal of Engineering & Science Research

a closed curve or a “fractal” in which case the system exhibits chaos. Chaos is unpredictable and random behavior occurring in a system that should be governed by deterministic laws. 3. ERGODIC HYPOTHESIS It is now appropriate to explain the ergodic hypothesis in physics in most simple words so that its use as analogy with democracy will be understood. According to this assertion made by Ludwig Edward Boltzman (1844- 1906), if you consider a gas consisting of particles and enclosed in specific volume each particle will pass through all the available phase space in course of time or in other words the probability that all the particles during the course of movements will pass through a particular point in space is finite. The ergodic hypothesis is the basis of statistical mechanics. Physical systems for which the ergodic hypothesis is valid are called ergodic systems. More precisely, in the classical mechanics of equilibrium systems, the ergodic hypothesis is the assertion that the time over age of the functions that depend on the coordinate and momenta of all the particles of a system in a phase space are equal to the statistical overages over a uniform distribution of phase points in a thin energy layers near a constant energy surface. The time averages are taken over the trajectory of the system as represented by points in the phase space. A uniform distribution of phase points in a thin energy layer near a constant –energy surface is called a Gibbs microcanonical distribution. In quantum statistical mechanics, the ergodic hypothesis is the assertion that all states in a thin energy layer are equally probable. Thus the ergodic hypothesis is equivalent to the assertion that a closed system can be described by a Gibbs canonical distribution and a Gibbs grand canonical distribution can be obtained on the basis of the microcanonical distribution. Mathematical ergodic theory studies the condition under which the time overages for dynamic systems are equal to the statistical averages. Ergodic hypothesis of this type were proved by the American Scientists G. Birkhoff and J. Von Neuman. According to the ergodic theorem of von Neumann, a system is ergodic when an energy surface cannot be divided into finite regions such that, if the initial phase point is located in one such region, the whole trajectory of the system remains entirely within that region. This property is called metric intransitivity. To prove that real systems are ergodic is a very complicated and as yet unsolved problem. Let us give an acceptable definition of ergodic hypothesis. It is a hypothesis in which concerns phase space. If a system of N-atoms or molecules is given by a point in 6N dimensional phase space with qi, representing coordinates and pi representing momenta. Taking the energy E to be constant the representative point in phase space describe an orbit the surface E (qi pi) =C, where C is a constant. The ergodic hypothesis states that the orbit of the representative point in phase space eventually goes through all point on the surface. 4. THE ANALOGY In the earlier section we have introduced the topic of ergodic hypothesis of statistical mechanics for our discussion with a non physics topic like democracy. It may also be thought as a view point for understanding ergodic hypothesis via analogies. It is worthwhile to cite few references [1-5] where ergodic hypothesis has been discussed and described in detail. For convenience we prepare a table (Table 1) which compares the parameters appearing in ergodic hypothesis and in the topic of democracy. Though we have considered six topics in both the cases for consideration of comparison, the topic No 3 is the central idea behind all the comparative studies. In the ergodic hypothesis all the microstates are equally probable after a sufficiently long period of time. Nobody has proved this assertion. In the domain of democracy we find that all the votes which are cost for a particular candidate in an election have equal values. Candidates are elected only through majority of the votes obtained. This is concerned with numbers only. One cannot distinguish between a vote cast by a prime minister and a rickshow puller. This appears to be paradoxical at first, but democracy has been proven as one of the most uniform system of government at any time.

J. Saikia et. al., / International Journal of Engineering & Science Research

Table 1: A Comparative analysis of the parameters appearing in ergodic hypothesis and democracy S. No. Ergodic Hypothesis Democracy 1 Eargodic hypothesis is the assertion made by L. The term[6] is derived from the Greek Boltzmann in the 1870’s word demokratia which was caried from demos (people) and jartos (rule) in the middle of the 5th century BC to denote the political systems then existing in some Greek city-states, notably Athens. This so- called democracy in Athens and Rome may be considered as precursors of modern democracies. 2 Ergodic hypothesis is applicable to microscope Democracy is applicable to macroscopic systems consisting of atoms, molecules and systems comprising of human beings subatomic parties. within specific age group. 3 In ergodic hypothesis all accessible microstates are In democracy [7], in general, all the equally probable over a long period of time. eligible voters cast their votes which have equal weightages. 4 Order disorder: Order disorder: As the Universe evolves in time (space-time) it Democracy is a system of government of will tend to move around phase space ergodically the whole population of a state typically (5). At a later time there is high probability that the through elected representatives. This is universe will be found in a state of disorder or presumably a state of high order and hence higher entropy because such states occupy most of a state of low entropy. If is reasonable to the phase space. This is generally accepted that the believe that at a later time there is high Universe started out in a state of high order. probability that a country (democratic) will be found in a state of disorder or higher entropy

Perhaps this justifies re-election after a specific time internal to maintain order. 5 In a gas system all the accessible microstates are In democracy all the eligible voters have equally probable after a ling interval of time. equal value. Though this is true it is seen These states cannot be manipulated. that voters may be manipulated by using many human factors[8]. 6 Maxwell-Boltzman distribution of speeds in gases, Democracy is represented by various Bose-Einstien distribution and Fermi-Dirac bodies like Panchayat, Assembly, distribution are all connected with ergodic Parliament etc. hypothesis. 5. SUMMARY AND CONCLUSION From what has been discussed in earlier sections it is appropriate to make a summary and conclusion of our work. We have made a comparative study between ergodic hypothesis in physics and the democracy. It may be noted that in many ways the phenomenon of democracy is analogous to ergodic hypothesis. Ergodic hypothesis is the basis of statistical physics while democracy is the basic of uniform government. We have selected six topics of interest for our comparative study between ergodic hypothesis and democracy. The central tenet of our comparison is that in ergodic hypothesis all accessible microstates are equally probable over a long period of time. In democracy all the eligible voters cast their votes which have equal values. The compressions which are have tabulate are expected to help us to understand ergodic hypothesis from analogy. The analogy has been male for the first time. REFERENCES [1] Arnold VI, Avez A. Ergodic problems of classical Mechanics, New York, 1968. [2] Zubarev DN. The great Soviet Encyclopedia, 3rd Edition (1970-1979). The ale Group. Inc. [3] Uhlenbeek G, Lektsii FJ. Po statisticheskoi mektronike, Moscow 1965;126-130. (Translated from English)

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[4] Tar D. Osnovania statisticheskoi Mechanik, Uspekhi fizicheskikh nauk. 59(4) and 60(1) (Translated from English). [5] Hawking SW. Phys. Rec. D 1985; 32(10): 2489. [6] Wilson NG. Encyclopedia of ancient Greece, New York, Routledge 2006; 511. [7] Dahl R. Democracy and its Critics. New Haven, Yale University Press, 1989. [8] Putnam R. Making Democracy Work, Princeton University Press, 2001.