A Quantum Shuffling Game for Teaching Statistical Mechanics

A Quantum Shuffling Game for Teaching Statistical Mechanics

P J Black, P Davies, J M Ogborn

1971

published in American Journal of Physics

Jon Ogborn (Chelsea College, University of London) and Paul Black (Physics Department, Birmingham University) collaborated with Bob Hopgood of the Atlas Computer Laboratory (ACL). This was the first set of commercial films made on ACL's SC4020 microfilm recorder starting in 1968.

• I. INTRODUCING IRREVERSIBLE PROCESSES
• II. THE QUANTUM SHUFFLING GAME
• III. COMPUTER FILMS
• IV. INTRODUCING ENTROPY
• V. JUSTIFYING THE GAME
• VI. FURTHER DEVELOPMENTS
• ACKNOWLEDGMENTS
• APPENDIX I
• REFERENCES

I. INTRODUCING IRREVERSIBLE PROCESSES

The basic strategy of an approach in which this game could play a part would be to draw attention to irreversible processes in general and to show that the operation of blind chance can account for them. In particular two types of example, involving diffusion and the flow of heat from hot to cold, would be stressed; development of the latter to establish the concept of temperature, via the zeroth law, is straightforward, but to show it as an example of the operation of the laws of chance and to develop the concept of entropy is not simple.

The diffusion type of problem can be investigated by analytic methods or by use of analog games. The arguments involve ideas like more probable, in a greater number of ways, the more likely thing will happen, and an approach depending solely on algebraic statistics may not help to establish the significance of these ideas. Simple games played with dice and tokens moved between two halves of a box can play a valuable role in establishing them for diffusion problems and in giving students a familiarity by direct contact with the laws of chance in operation. The development could also be aided by computer simulation of the games. [2] [3]

The game to be described here might be the next exercise after a diffusion game and is concerned with showing that random interchange of energy amongst distinguishable elements gives rise to an exponential distribution, the exponent of which depends on temperature.

II. THE QUANTUM SHUFFLING GAME

The simplest system to investigate is an Einstein model of a crystal. In the game, the crystal is represented by a regular array of (distinguishable) sites, each of which has a set of vibrational energy levels which are uniformly spaced. Zero-point energy is neglected, the possible vibration energies of each and every site are assumed to be O, ε, 2ε, 3ε, ···, and sites are assumed to be loosely coupled so that they can interchange energy, in quanta of size ε, although this is assumed to have no effect on their independent energy levels. The game can be played in its simplest form by presenting students with a flat square mesh of 6 × 6 sites and giving them a set of identical tokens, representing energy quanta to be laid out on the sites; 36 tokens laid out with one on each site might be a suitable starting pattern. A throw of a pair of six-sided dice generates a pair of random numbers which are used as coordinates to select a site at random. An energy quantum is taken up from the selected site and a second throw of the dice selects a site to which this quantum is moved. This pair of throws completes a move. The class then tries to find out the effect of making a long sequence of random moves, and are shown how to represent the state of the system by an energy distribution histogram showing the numbers of sites with zero, one, two, ... etc., quanta. A problem soon arises when the first throw of a move selects a site containing no quanta; this is regarded as a completed move and students are advised to go on to start the next move. Moves of this type only present a problem if time averages over a sequence of moves are taken; the problem is avoided if students observe their own instantaneous distributions and form averages by combining a number of such distributions produced by separate students (see Section V).

After about 100 moves a graded distribution is found, approximating roughly to the ideal exponential distribution which for 36 sites and 36 quanta would give 18 at zero, 9 at ε, 4.5 at 2ε, 2.25 at 3ε, and so on. The result usually comes as a surprise to a class, and questions about what will happen next can lead to further moves and to the experience that the system fluctuates about the ideal distribution. The distribution for one 36-site board fluctuates far from the ideal, but the average of about 10 such boards is more stable and more easily recognized as exponential.

III. COMPUTER FILMS

Further exploitation of the game requires the use of computer results, for although the direct experience of playing with a small crystal is of great importance at the outset, a great deal can be accomplished subsequently if the speed and size of the game can be increased and if a class can accept that the computer is performing the same operations for them. The sequences described below have been presented in computer-made films which were prepared at the Atlas Computer Laboratory [4] and which will be published. [5] The sequences are as follows:

1. Nine-hundred quanta are assigned, at one per site, to a 30 × 30 array, and quanta shuffled at random for 10,000 moves. Examination of the energy distribution histogram shows that it fluctuates, without any marked change, after about the first 1000 moves; the mean distribution shows an approximately 2:1 ratio between the numbers having n and n+1 quanta. This is as it should be since in these statistics, with N sites and a total of q quanta, the ratio should be (N+q)/q when N and q are large. If N = q as here, the ratio is 2. (Film 1)
2. This is the same as (a) except that for the starting distribution 300 sites have no quanta, 300 one quantum each, and 300 two quanta each. The final steady state is the same; this serves to show that the result does not depend on the initial distribution. (Film 2)
3. The array is altered to a 25 × 25 set with one quantum each, and this gives a distribution of the same shape with the same grading ratio, showing that with a different number of sites and quanta the grading ratio is the same. (Film 3)
4. In the fourth film the system used for (a) or (b) is used again but attention is concentrated on one particular site and the number of quanta on it is recorded after every move (most moves of course leave this particular site unchanged). A time-average distribution is built up for the one site which yields a graded distribution, although after 4 × 10⁵ moves (the limit of the film) it has not reached the ideal ratio (2:1), for which about 2 × 10⁵ moves would be needed. The purpose of this film is discussed below. (Film 4)
5. An array with 900 sites but with only 300 quanta is used, and this gives a distribution with a ratio of 4:1 between numbers having n and n+1 quanta; a colder system has a more steeply graded distribution. (Film 5)
6. Two arrays, each of 900 sites, are set alongside and shuffled independently, one having 900 quanta the other 300, and these lead to 2:1 and 4:1 distributions; the first array is hot and the second cold. They are then placed in contact and shuffled as one system of 1800 sites. The quanta become uniformly distributed and the distributions of the two halves alter until they both show a ratio of about 2.5:1. This is a simple demonstration of the operation of random processes in giving a net heat flow from hot to cold until thermal equilibrium is reached. The computer is not instructed to transfer quanta, it just happens because the uniform temperature state can be realized in a greater number of ways than the initial non-uniform state. (Film 6)

IV. INTRODUCING ENTROPY

The game and film should have established the link between heat flow, temperature, and the operation of blind chance and this appreciation is a good basis for developing statistical thermodynamics. Only a very brief description of the teaching which can be developed from these films will be given here. Film (d) can be used in a microcanonical ensemble argument: Any one atom has (say) one quantum twice as often as two, so the rest of the crystal must have twice as many ways if it has an extra quantum, the factor two being its grading ratio. With a four to one ratio, it would have four times as many ways for an extra quantum, so quanta will tend to transfer from a 2:1 crystal to a 4:1 crystal. If only a few (say p) quanta are transferred, the energy gained is pε, the number of ways (W) goes up by a factor depending on p and on the steepness of the distribution, so δ ln W is proportional to p, i.e., δ ln W depends on δQ, where δQ=pε. The factor depending on the gradient of the distribution indicates that the change for given δQ is larger for cold than for hot crystals, so a temperature scale can be defined by saying δ ln W = δQ/kT, k being introduced as a scale factor. From this result much of the standard work involving the use of the concept of entropy, understood from a microscopic standpoint, can be developed. Good simple arguments are presented in Bent.[6]

V. JUSTIFYING THE GAME

The conventional way of establishing a Boltzmann distribution for an Einstein model of a crystal is to assume that all possible microstates are equally likely and to compute either that the mean of all possible microstates is an exponential distribution or that the most probable distribution is exponential. [7] [8] The arguments developed in this approach show that the total number of ways (W) for N sites and q quanta is given by

W = (N + q - 1)! / (N - 1)! q!           (1)

It also follows from the analysis that the ratio between the number of sites having n quanta and the number having (n+1) quanta is (N+q)/q, and that the fraction of the sites having zero quanta is N/(N+q); the results given by the computer film for the game are in agreement with these last two predictions.

It is not clear a priori whether the shuffling game as defined should lead to the Boltzmann distribution; if it does so it is a new way of establishing that distribution because the initial assumptions that it requires are different from those adopted in conventional proofs. In the conventional treatment for an Einstein model of a crystal, the starting assumption of equal a priori probability for all states is not usually supported by any arguments to show how a crystal could have access to all of these states. The assumption of equal a priori probability of microstates is usually justified by results; predictions based upon it agree with experiment.

One might take a similar view of the game proposed and justify its rules by their outcome, without linking the rules to any other proposition, such as that of equal a priori probability. Nevertheless, the conventional theoretical treatment exists and works, and the mathematical question of whether the average stationary state of a sequence of random moves, made according to the rules of the game, does give the same result as the conventional argument is an interesting one. A proof that the rules of the game are equivalent to an assumption of equal a priori probability is presented in Appendix I. It is essential to this proof that null moves (i.e., moves in which the site selected for removal of a quantum has no quanta to donate) are regarded as making no change but are reckoned as completed moves. This justifies the rule about these moves quoted in Section II. If, as in the films, a time average distribution over a long sequence of moves is taken, then states with a large number of zeros persist for many moves and so have more weight in the time average, if the null moves are not ignored.

At an elementary level, the advantage of treating the rules of the game as arbitrary postulates is that they may seem to a student simpler or more intelligible, and perhaps more natural, than the assumption of equal a priori probability. However the assumptions of the game, that quanta are exchanged one at a time and that the distance separating atoms in the crystal has no bearing on the likelihood of an exchange of energy, are not easily justified. We have not investigated the properties of a game in which the second part of a move is restricted to one of the near neighbours of the site from which a quantum is taken up in the first part; a three-dimensional model and a cyclic boundary condition would presumably be needed for such a game. At a more sophisticated level, the distinguishable elements could be regarded as the normal modes of a crystal instead of the atomic sites: Free interchange would then be more acceptable, but the equal spacing of the levels would not.

VI. FURTHER DEVELOPMENTS

In addition to the possibility discussed in the previous paragraph, other developments of the game are possible and ought to be investigated. The present form of film (f) is probably an optimum choice for a first teaching approach but needs more sophisticated examples if it is to establish that heat flow is more than simple diffusion and that temperature is not the same as the mean number of quanta per crystal site. To generalize the results in this way would require games involving sites with different energy level spacings.

A further area we have not fully explored is that of modifying the rules so that the game gives Bose-Einstein or Fermi-Dirac statistics; the rules described above yield Maxwell-Boltzmann statistics. Since the single assumption of equal a priori probability of microstates, together with specifications of what constitutes a distinct microstate, serves to generate these different statistics, it may be thought that the conventional approach would, in these extensions of statistical mechanics, have advantages. It may be so; our view would depend on whether the necessarily modified rules for games generating other types of statistics seemed capable of natural and simple interpretation in terms of plausible processes within a material or whether they seemed arbitrary.

ACKNOWLEDGMENTS

We would like to thank F R A Hopgood of the Atlas Computer Laboratory, who produced the films, several colleagues in the physics department of the University of Birmingham for valuable discussions, Professor D J Millen of University College London who first drew our attention to this type of game, and Professor K W Keohane, of Chelsea College London, coordinator of the Nuffield Foundation Science Teaching Project, for advice and encouragement.

APPENDIX I

Suppose the system has N sites among which q quanta are distributed. Any one state of the system can be represented by specifying the numbers, X _l of quanta on each site l in the list (X₁ , X₂ , ... X_N ) where

ΣX_l = q for l = 1,N

For example, with N = 4, q = 6, some possible states are 6000, 4200, 0033, and 1122. The sites are distinguishable so 6000, 0600, 0060, 0006 are counted as separate states, but the quanta are identical so no distinction is made between the six ways of choosing the single quantum in the state 5100.

Suppose that r ₀ of the X's are zero, r ₁ are unity, and so on, so that

Σi r _i = q for i=0 to q

Each set of r's corresponding to a possible distribution may be given by several states. For example, r ₀=3, r ₁=r ₂=r ₃=r ₄=r ₅=0, r ₆=1 can be given by the states 6000, 0600, 0060, and 0006. In the shuffling game, a sequence of states is produced by the moves. For given N and q, there are W possible states where

    (N+q-1)
W = (     )
    (  q  )

Let the states be labelled s ₁, s ₂, ···, s _W and let P _{s j} , be the probability that the system is found in s _j on some initial observation of the system. The probability that a move in the game takes the system from s _j to s _i depends only on the states s _i and s _j and can be represented by p _{i j} , where

Σp _{i j} = 1 for i=1 to W

The total probability P _{s i} ^' of being in state s _i after one move is related to the previous P _{s i} ...P _{s W} by

P _{s i} ^'=Σ p _{i j} P _{s i} for j=1 to W          (2)

The shuffling game therefore corresponds to a statistical process known as a Markov chain with a finite number of states. Such processes are discussed, for example, in Feller.[9] Further, it is clear that starting from any state s _i in the game it is possible to reach any other state s _j in a finite sequence of moves. Such a Markov chain is said to be irreducible. For such irreducible chains, it is known that the probability distribution (P _s1,...,P _sW ) tends with an increasing number of moves to a unique, stable or stationary distribution such that the stationary probabilities (P _s1,...,P _sW ) satisfy Equation (2) with P _{s i} ^'=P _{s i} .

For the crystal model used here with the number of quanta per site taking possible value between 0 and q the physical distribution of the r _i 's is the Boltzmann distribution as previously stated (see, e.g., Gurney [8], Sherwin [9]). This corresponds to having equal stationary probabilities for the states or

P _s1=P _s2...=P _sW =W ^-1          (3)

If the game produces all possible states with equal probability, then it will give the Boltzmann distribution for the r _i 's. There is no a priori physical reason why the game should produce the same results as a crystal. The crystal might follow a Boltzmann distribution because its quantum shuffling is played by different, perhaps more complicated, rules. However the game produces the required Boltzmann distribution for the r _i 's if Equation (3) is the unique solution of Equation (2) for the game and this is now proved.

Consider all the possible moves which could lead to a particular state (X ₁,X ₂,...X _N ). Any one parent state can be altered by N ² different moves so that a given move has probability 1/N ². Consider first the moves involving a change of state from s _j to s _i (j≠i). There is only one such move connecting s _j to s _i with probability 1/N ² for i≠j. Two types of such moves are possible:

(a) The first type are moves for which one of the (N-r ₀) nonzero sites of s _i receives a quantum from one of the (N-r ₀-1) donor sites of s _j, i.e., from one of these sites in s _j, which end up as nonzero in s _i . For example if the final state was 0312, there would be one move from each of 0222, 0213, 0402, 0303, 0411, and 0321 which could lead to the final state. Thus there are (N-r ₀)(N-r ₀-1) moves each with probability 1/N ².

(b) The second type are moves in which one of the sites which end up as zero in s _i is nonzero in s _j and becomes zero by donating a quantum. Examples might be moves taking 1212, 1311, 1302 to the state 0312. There are r ₀ ways of choosing the donor site and (N-r ₀) ways of choosing the recipient. Thus there are r ₀(N-r ₀) moves of this type each with probability 1/N ².

Lastly, there are identity moves leaving sites as they were before the move. For each move such that the donor is one of the (N-r ₀) nonzero sites, there is only one identity move. For each move such that the donor is chosen as one of the r ₀ zeros, there are effectively N recipients for each donor. Thus the total number of identity moves is (N r ₀+N-r ₀) each with probability 1/N ².

It follows that

Σp _ij for i=1 to W= (1/N ²)[N r ₀+N-r ₀ +(N-r ₀)(N-r ₀-1) +r ₀(N-r ₀)]=1.

Now (3) specifies a solution of (2) if Σp _ij for i=1 to W=1

and the solution will be unique. This has been proved for the shuffling game which therefore gives the required Boltzmann distribution.

REFERENCES

(1) This course is in the final stage of trials; it is scheduled for publication by Penguin early in 1972. For a brief general account see P J Black and J M Ogborn, Phys. Bull. 21, 301 (1970).

(2) F. Reif, Statistical Physics Berkeley Physics Course. Volume 5 (McGraw--Hill, New York, 1967).

(3) Physical Science Study Committee, College Physics (Raytheon, Boston, 1968).

(4) Atlas Computer Laboratory, Chilton, Didcot, Berkshire, Eng. Films were programmed and produced by F R A Hopgood, using the Atlas Computer and a Stromberg-Carlson 5C-4020 high speed CRT recorder.

(5) Penguin Books Limited, Harmondsworth, Middlesex, are producing these films as a 16-mm film.

(6) H A Bent, The Second Law (Oxford U. P., New York, 1965).

(7) R W Gurney, Introduction to Statistical Mechanics (McGraw -Hill, New York, 1949).

(8) C W Sherwin, Basic Concepts of Physics (Holt, Rinehart and Winston, New York, 1961).

(9) W Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1950), Vol. 1.

FILMS

Film 1

Film 2

Film 3

Film 4

Film 5

Film 6