EEconophysics as conceived by Meghnad Saha
Bikas K. ChakrabartiSaha Institute of Nuclear Physics, Kolkata 700064S. N. Bose National Centre for Basic Science, Kolkata 7000106Economic Research Unit, Indian Statistical Institute, Kolkata 700108
Abstract:
We trace the initiative by Professor Meghnad Saha to develop a(statistical) physics model of market economy and his search for the mechanismto constrain the entropy maximized width of the income distribution in asociety such that the spread of inequality can be minimized.
I. Introduction:
Professor Meghnad Saha had been a great physicist andperhaps an even greater social scientist and reformer. It is this second as-pect, which is not analysed often or highlighted much in the literature abouthim. In this brief article, I would like to discuss his science-based attempts tomodel social dynamics and about his inspiring attitude towards social sciencein general.
II. Saha’s science-based approaches for social problems:
There areseveral writings about him which talks about Professor Saha’s deep interestand involvement in matters of our day-to-day social concern. Let me quotefrom a book and a recent article on Saha, as examples: Dilip M. Salwi, inhis book titled ‘Meghnad Saha: Scientist with a Social Mission’ [1] writes “ ...Meanwhile in 1943, Damodar river went into spate, its flood waters even surroundedKolkata and cut it off from rest of India. ... Saha took upon himself the taskof studying the flood problem of Damodar river in totality because it also causedsoil erosion and siltation. In addition to the study of the topography of the region,the annual rainfall at various spots, etc., he set up a small hydraulic laboratory inthe college to simulate the actual conditions and understand the problem. He alsoconducted a first-hand survey of the region and even studied the various flood control a r X i v : . [ q -f i n . GN ] A ug easures of river systems of the western world. On the basis of his painstakingresearch, he wrote a series of articles in his own newly launched monthly Scienceand Culture. ... To sensitize his colleagues, friends, and students to the menace offloods, he also set up a model of Damodar river system in the corridor of ScienceCollege! ”. Pramod V. Naik in his article [2] writes “ ... Meghnad Saha (1893-1956) ... was of the firm opinion that in a country like India, the problems offood, clothing, eradication of poverty, education and technological progress can besolved only with proper planning, using science and technology. ... While thinkingabout various issues of national importance, Saha realized the need for a scientificperiodical, ... and founded a monthly titled Science and Culture. ... The rangeof topics in editorials and general articles was amazing; for example, the need for ahydraulic research laboratory, irrigation research in India, planning for the DamodarValley, ..., public supply of electricity in India, national fuel policy, ... mineralsources and mineral policy, problem of industrial development in India, automobileindustry in India, industrial research and Indian industry, industrial policy of thePlanning Commission, ... and so on. Every article shows his inner urge for the goalof national reconstruction. ... show his direct or indirect role as a social thinker andreformer. ”. See also the paper by Vasant Natarajan is this special issue. III. Saha and Srivastava’s Kinetic Theory for ideal gases with ‘atoms’and ‘social atoms’:
No wonder, Professor Saha had thought deeply aboutthe scientific foundation of many social issues. In particular the omnipresenceof social (income or wealth) inequalities in any society throughout the historyof mankind must have caught his serious attention. Like all the philosophersthrough ages, and also as a born socialist, he must have been pained to seesocial inequalities cutting across the history and the globe at any point oftime. Yet, his confidence in science, statistical physics in particular, perhapsconvinced him that ‘entropy maximization’ principle must be at work in the‘many-body’ system like the society or a market, and some amount of inequal-ity might be inevitable! One Robinson Crusoe in an island can not developa market or a society. A typical thermodynamic system, like a gas, containsAvogadro number ( N about 10 ) of atoms (or molecules). Compared to this,the number ( N ) of ‘social atoms’ or agents in any market or society is of coursevery small (say, about 10 for a village market to about 10 in a global market).Still such many-body dynamical systems are statistical in nature and statisti-cal physical principles should be applicable. In the famous text book ‘Treatiseon Heat’ [3], written together with Professor Biswambhar Nath Srivastava, thestudents were encouraged, in the section on Maxwell-Boltzmann distribution2n kinetic theory of ideal gas, to think about applying kinetic theory to themarket and find the income distribution in a society, which maximizes the en-tropy (for stochastic market transactions). It may be noted, this point was notput at the end of the chapter. It was put right within the text of the sectionon energy distribution in ideal gas. It provokes the student to search for theequivalent money or income (wealth) distribution in a stochastic market whenmoney is conserved in each of the transactions. S ca nn e d by C a m S ca nn e r Figure 1: Photocopy of the pages 104 and 105 of Ref. [3], where theauthors urge the students to use kinetic theory to get the Gamma-like incomedistribution indicated in Fig. 6 of page 105.One can present the derivation of the energy distribution in an ideal gasin equilibrium at a temperature T as follows: If n ( (cid:15) ) represents the num-ber density of particles (atoms or molecules of a gas) having energy (cid:15) , then n ( (cid:15) ) d(cid:15) = g ( (cid:15) ) f ( (cid:15), T ) d(cid:15) . Here g ( (cid:15) ) d(cid:15) denotes the denotes the ‘density of states’3iving the number of dynamical states possible for free particles of the gas, hav-ing kinetic energy between (cid:15) and (cid:15) + d(cid:15) (as dictated by the different momentumvectors (cid:126)p corresponding to the same kinetic energy: (cid:15) = | (cid:126)p | / w , where w de-notes the mass of the particles). Since the momentum (cid:126)p is a three dimensionalvector, g ( (cid:15) ) d(cid:15) ∼ | (cid:126)p | d | (cid:126)p | ∼ | (cid:126)p || (cid:126)p | d | (cid:126)p | ∼ √ (cid:15)d(cid:15) . This is obtained purely from me-chanics. For completely stochastic (ergodic) many-body dynamics or energyexchanges, maintaining the the energy conservation, the energy distributionfunction f ( (cid:15) )( ≡ f ( (cid:15), T )) should satisfy f ( (cid:15) ) f ( (cid:15) ) = f ( (cid:15) + (cid:15) ) for any arbi-trary (cid:15) and (cid:15) . This suggests f ( (cid:15) ) ∼ exp ( − (cid:15)/ ∆), where the factor ∆ can beidentified from the equation of state for gas, as discussed later (= κ T , κ de-noting the Boltzmann constant). This gives n ( (cid:15) ) = g ( (cid:15) ) f ( (cid:15) ) ∼ √ (cid:15)exp ( − (cid:15)/ ∆).Knowing this n ( (cid:15) ), one can estimate the average pressure P the gas exerts onthe walls of the container having volume V at equilibrium temperature T andcompare with the ideal gas equation of state P V = N κ T . The pressure canbe estimated from the average rate of momentum transfer by the atoms onthe container wall and one can compare with that obtained from the above-mentioned equation of state and identify different quantities; in particular, oneidenifies ∆ = κ T .Saha and Srivastava’s chosen example in this section (see Fig.1) indicatedto the students that since in the market money is conserved as no one can printmoney or destroy money (will end-up in jail in both cases) and the exchangeof money in the market is completely random, one would again expect, forany buyer-seller transaction in the market, f ( m ) f ( m ) = f ( m + m ), where f ( m ) denotes the equilibrium or steady state distribution of money m amongthe agents in the market. This, in a similar way, suggests f ( m ) ∼ exp ( − m/σ ),where σ is a constant. Since there can not be any equivalent of the particlemomentum vector for the agents in the market, the density of states g ( m )here is a constant (each value of m corresponds to one market state). Hence,the number density n ( m ) of agents having money m will be given by n ( m ) = Cexp ( − m/σ ), where C is another constant. One must also have (cid:82) ∞ n ( m ) dm = N , the total number of agents in the market and (cid:82) ∞ o mn ( m ) dm = M , thetotal money in circulation in the country. This gives, C = 1 /σ and σ = M/N , the average available money per agent in this closed-economy traders’market (as no growth, migration of labourers, etc. are considered). This givesexponentially decaying (or Gibbs-like) distribution of money in the market(unlike the Maxwell-Boltzman or Gamma distribution of energy in the idealgas), where most of the people become pauper ( n ( m ) is maximum at m = 0).They asked the students to investigate, what could make the distribution more4ike Gamma distribution, as seemed to them to be observed phenomenon inmost of the societies. Any understanding towards that would help to identifyand develop public policies to reduce the extent of inequalities in the society.It may be mentioned here that although the statistical aspects of societies,markets in particular, had been addressed in some speculative papers earlier,the 1931 book ‘Treatise of Heat’ [3] is the first ever text book addressing thequestion in the context of kinetic theory of gas and pleads for the search for asolution from statistical physics! IV. Saha’s Econophyics contribution as noted in some on-line en-clyopedias:
In the entry on ‘History of Indian School of Econophysics’, inthe online encyclopedia (Hmolpedia) [4] Libb Thims (2016) writes: “ ... In1931, Indian astrophysicist Meghnad Saha (1893-1956), an atheist, in his Treatiseon Heat, co-authored with B.N. Srivastava, explained the Maxwell- Boltzmann dis-tribution of molecular velocities according to kinetic theory in terms of the wealthdistributions in society [Saha, Meghnad and Srivastava, B. N. (1931), Treatise onHeat (pg. 105), The Indian Press Ltd. (1931)]. Indian physicist Prasanta Maha-lanobis (1893-1972), interested in Karl Pearson stylized biometrics, founded in 1931the Indian Statistical Institute for developing physical and statistical models for so-cial dynamics. In 1938, Saha began to form the Saha Institute of Nuclear Physicsin Kolkata, India. ”. The Wikipedia entry (June, 2018) on ‘Kinetic exchangemodels of markets’ [5] reads: “ ... Kinetic exchange models are multi-agent dy-namic models inspired by the statistical physics of energy distribution, which try toexplain the robust and universal features of income/wealth distributions. Under-standing the distributions of income and wealth in an economy has been a classicproblem in economics for more than a hundred years. Today it is one of the mainbranches of econophysics. ... In 1897, Vilfredo Pareto first found a universal featurein the distribution of wealth. After that, with some notable exceptions, this fieldhad been dormant for many decades, although accurate data had been accumulatedover this period. Considerable investigations with the real data during the last fifteenyears revealed that the tail (typically 5 to 10 percent of agents in any country) ofthe income/wealth distribution indeed follows a power law. However, the majorityof the population (i.e., the low-income population) follows a different distributionwhich is debated to be either Gibbs or log-normal. ... Since the distributions ofincome/wealth are the results of the interaction among many heterogeneous agents,there is an analogy with statistical mechanics, where many particles interact. Thissimilarity was noted by Meghnad Saha and B. N. Srivastava in 1931 [Saha, M. &Srivastava, B. N., A Treatise on Heat. Indian Press (Allahabad; 1931). p. 105], nd thirty years later by Benoit Mandelbrot . In 1986, an elementary version of thestochastic exchange model was first proposed by Jhon Angle [Angle, J. (1986). Thesurplus theory of social stratification and the size distribution of personal wealth,Social Forces, (1986), pp. 293-326]. ” V. Brief discussions on recent developments:
Saha and Srivastava hadbeen in search of the mechanism that will ensure the initial fall (as in theMaxwell-Boltzmann distribution n ( (cid:15) ) of energy for the Newtonian particles inthe gas, due to the particle momentum count of the density of states) in theGibbs-like money distribution n ( m ) in the market . They inspired the stu-dents to investigate the plausible reasons for this ‘Gamma’ distribution likeinitial fall (from the scenario where most people are pauper or n ( m ) is maxi-mum at m = 0), which they rightly conjectured to be the realistic form of themoney distribution in any society. There was, however, another aspect of thedistribution n ( m ) in any society. This was first observed and formulated in1896 by Vilfredo Pareto (Engineer cum economist/sociologist of the Polytech-nic University of Turin), known as Pareto law [6]. This law states that for thenumber of truly rich people in any society n ( m ) does not fall off exponentially(as in the kinetic theory model indicated above), rather fall off with a ‘fat tail’having an ‘universal’ power law decay: n ( m ) ∼ m − α , with α -value in the range2 - 3.The market model considered above by Saha and Srivastava could at bestbe described as a trading market (with fixed value of N and M ) as it does notaccommodate in this limit any growth, industrial development or migrationof labours etc. If one consults a standard economics text book and looks forthe discussions on trading market, one can not miss the discussions on ‘savingpropensity’ ! This quantity characterizes the agents or traders by the fractionof their money holding at that instant they save before going in for a trade. Mandelbrot, B. B., The Pareto-Levy law and the distribution of income. InternationalEconomic Review. (1960) pp. 79-106; Indeed he wrote: There is a great temptationto consider the exchanges of money which occur in economic interaction as analogous tothe exchanges of energy which occur in physical shocks between molecules. In the loosestpossible terms, both kinds of interactions should lead to similar states of equilibrium.That is, one should be able to explain the law of income distribution by a model similarto that used in statistical thermodynamics: many authors have done so explicitly, and allthe others of whom we know have done so implicitly. For a recent account on the extent and consequences of this Gibbs-like distribution inmonetary econophysics, see: V. M. Yakovenko and J. Barkley Rosser, Statistical mechanicsof money, Reviews of Modern Physics, , 1703-1725 (2009) n (0) will drop down to zero from its maximum for any non-zero savingpropensity of the agents, in the steady state. The most probable income peragent in the market will shift from zero to a value dependent on the savingpropensity of the agents. It also suggests immediately that in a market withpeople having different or random values of saving propensity, there would bea net flow of money towards people with higher saving propensities and in thesteady sate, a robust power law tail for the distribution, n ( m ) ∼ m − ( m →∞ ), will ensue for most of the non-singular probabilities of saving propensityamong the population [7]. Many important developments and modificationson this kinetic exchange model, pioneered by Saha and Srivastava in [3] (seealso [7]), have taken place since then, both in the mathematical physics (seee.g., [8]) and in economics (see e.g., [9]). Concluding Remarks:
Apart from being one of the pioneering astrophysi-cists of our time, Professor Meghnad Saha had been a passionate and enthusi-astic thinker in social science. His deep conviction about scientific approaches7o social phenomena convinced him about the applicability of the laws physics,of statistical physics in particular, to social sciences. In view of the inherentmany-body stochastic dynamical features of the markets, he realised that theentropy maximizing principle will not support a steady state narrow incomeor money distribution among the traders. As a diehard socialist, this con-clusion must have been quite painful to digest. He was thus in search of thetuning parameters for controlling the width of the money distribution n ( (cid:15) ),around the most probable value of income (ideally the average money peragent M/N in circulation in the market). Saha-Srivastava’s kinetic exchangemodel for a traders’ market suggested that most traders will beecome pauperin the steady state ( n ( m ) will be maximum at m = 0). This comes due to thelimitation of their model (where the social atoms can lose its entire accumu-lated money in the next trade with another, as in the case of real atoms inthe gas) and also such money distributions do not quite compare either withthe observational results in societies. They had, however, every confidenceon the students, whom they expected to think about it and find eventuallya proper solution. As mentioned above in section V, one such solution hadbeen achieved later by adding finite saving propensities of the social atoms ortraders in the Saha-Srivastava model, following the observations of economistsin the trading markets. This additional feature in the model naturally givesGamma-like income distribution with tunable dispersion of inequality near themost probable income and also induces the eventual Pareto-like slowly decay-ing numbers (fat tail distribution) of the ultra-rich people or traders in thesociety. Acknowledgement:
I am grateful to Abhik Basu, Arnab Chatterjee, KishoreDash, Amrita Ghosh, Libb Thims and Sudip Mukherjee for important com-ments, criticims and communications.
References: [1] D. M. Salwi,
Meghnad Saha: Scientist with a Social Mission , Rupa & Co.,New Delhi (2002), pp. 49-51[2] P. V. Naik, ‘Meghnad Saha and his contributions’, Current Science, Vol.811(1), 10 July (2016), pp. 217-218[3] M. N. Saha and B. N. Srivastava,
A Treatise on Heat.
Indian Press (Alla-habad; 1931), p. 105[4] L. Thims, ‘History of Indian School of Econophysics’, in free online Ency-clopedia of Human Thermodynamics (Hmolpedia; as of June 2018): [5] Wikipedia entry (as of June, 2018) on ‘Kinetic exchange models of markets’,in free online Encyclopedia: https://en.wikipedia.org/wiki/Kinetic_exchange_models_of_markets [6] V. Pareto,
Cours dEconomie Politique , Droz, Geneva (1896)[7] B. K. Chakrabarti, A. Chakraborti, S. R. Chakravarty and A. Chatterjee,
Econophysics of Income & Wealth Distributions , Cambridge University Press,Cambridge (2013)[8] L. Pareschi and G. Toscani,
Interacting Multiagent Systems , Oxford Uni-versity Press, Oxford (2014)[9] M. Shubik and E. Smith,