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The Boltzmann Distribution

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The Boltzmann Distribution THE BOLTZMANN DISTRIBUTION ZHENGQU WAN Abstract. This paper introduces some of the basic concepts in statistical mechanics. It focuses how energy is distributed to different states of a physical system, i.e. under certain hypothesis, it obeys the Boltzmann distribution. I will demonstrate three ways that the Boltzmann distribution will arise. Contents 1. The Motivating Problem 1 2. Boltzmann Distribution Arises from the Principle of Indifference 4 3. Validity of the Principle of Indifference 7 4. Boltzmann Distribution Arises as the Maximal Entropy Distribution 9 5. Application to Particles in the Box 12 6. Application to Gas Particles under Gravity 13 7. Boltzmann Distribution Arises as the Equilibrium Distribution 13 Acknowledgments 14 References 14 1. The Motivating Problem Consider the following game. Suppose there are N people and everyone has some cash. For simplicity, each person's cash is a natural number. Now imagine the following process. At each second we randomly uniformly pick a person A. If A has positive amount of cash, then we randomly uniformly pick a person B(it could be the same person) and transfer a dollar from A to B. Keep this game running for a sufficiently long period of time, will the distribution of wealth among this group of people reach an equilibrium. If so, what will the equilibrium distribution be? The answer to the above question is affirmative. Let Xi denote the wealth of the i-th person. Let u denote the average wealth so that Nu is a positive integer. The game can be modeled by a Markov chain where the state space is n X S = (X1; :::XN ): Xi 2 Z>0; Xi = nu i=1 Theorem 1.1. The Markov chain is transitive and periodic. Proof. Starting at any initial state X = (X1; :::; XN ), it is possible to transfer all the cash to the first person and thus reach state (Nu; 0; :::; 0) and vice versa. Thus Date: 2017, July, 13. 1 2 ZHENGQU WAN the Markov chain is transitive. Since it has positive probability for the state X to remain unchanged, the Markov chain is periodic. Theorem 1.2. The transition probability from any state to any of its neighboring 1 states is N 2 . Thus the stationary distribution of this Markov chain is the uniform distribution π on S. Proof. For each state X = (X1; :::; XN ), to transition to its neighboring state (X1; :::; Xi − 1; :::; Xj + 1; :::; XN ) (assumming Xi > 0), we have to pick i-th person first and then pick j-th person. Since there are N people in total, the probability 1 is N 2 . For any pair neighboring states X and Y , since we have equal transition probability from X to Y and from Y to X, the uniform distribution on S is sta- tionary. Since the Markov chain is irreducible and aperiodic, any initial probability distri- bution of the Markov chain will converge to the uniform distribution π. Therefore, after the game runs for a sufficiently long period of time, we are equally likely to observe any state in S. To describe the the distribution of wealth, we just have to know how many (in term of expected value) people have cash w for each w. Notice N N X X E[#People with wealth w] = E[ 1fXi = wg] = P r[Xi = w] i=1 i=1 So we just have to know the distribution of each Xi. We can explicitly compute P r[Xi = w] #Ways to distribute (Nu − w) wealth to (N − 1)people = #S Nu − w + N − 2Nu + N − 1 = N − 2 N − 1 (Nu − w + N − 2)! (N − 1)!(Nu)! = ∗ (N − 2)!(Nu − w)! (Nu + N − 1)! We can use Stirling's approximation to obtain the asympototic behavior of the above expression, but then we would trap ourselves in massive computations. In- stead, we will study the continuous analogue of this problem in the next section, which is easier to deal with. If the amount of wealth being transferred each time is some small number 2−m instead of 1 dollar, we will obtain the uniform distribution on the finer grid on the simplex. n −m X S = (X1; :::XN ): Xi 2 2 Z>0; Xi = Nu i=1 When m gets larger, the equilibrium distribution converges (weak* convergence) to the uniform probability distribution on the (N − 1)-dimensional simplex N X S = (X1; :::; Xn): Xi 2 R>0; Xi = Nu i=1 THE BOLTZMANN DISTRIBUTION 3 To obtain the probability distribution of individual Xi, we observe that, for t 2 t [0; Nu], the region fXi > tg is still a simplex but with size (1 − Nu ) times as large as S. Therefore t N−1 P r[X t] = 1 − i > Nu Thus the density function is thus N − 1 t N−2 ρ (t) = 1 − ; t 2 [0; Nu] N Nu Nu Send n ! 1, we obtain 1 t ρ = exp(− ); t 2 [0; +1) 1 u u This is the famous Boltzmann Distribution in statistical mechanics, which tells us that it is less likely to find the system in higher energy states, with the prob- ability being inverse proportional to the exponential of the energy. In this example, the wealth is the analogue of energy in physics and we see that the number of people with certain wealth decreases exponentially with the wealth. Another important feature is that these Xi are Asymptotically Independent, i.e.the joint distribution of the first k variables (X1; :::; Xk) tends to be independent when n ! 1. Such feature is analogous to the fact in real world that, position and momentum of different gas molecules can be regard as independent random variables, in spite of their collisions and interactions. To prove the asymptotical independence, we first observe that, for tj > 0 and P j tj 6 Nu, the region fX1 > t1; :::; Xk > tkg is still a simplex but with size t1+:::tk (1 − Nu ) times as large as S. Therefore t + ::: + t N−k−1 P r[X t ; :::; X t ] = 1 − 1 k 1 > 1 k > k Nu So we obtain the joint density 1 (N − k − 1)! t + ::: + t N−2k−1 ρ (t ; :::; t ) = ( )k 1 − 1 k N 1 k Nu (N − 2k − 1)! Nu Send N ! 1, we obtain 1 t + ::: + t ρ (t ; :::; t ) = exp(− 1 k ) 1 1 k uk u Note k 1 t1 + ::: + tk Y 1 ti exp(− ) = exp(− ) uk u u u i=1 Thus we conclude as N ! 1, the individual wealth X1; :::; Xk tends to be inde- pendent, while each of them obeys the Boltzmann distribution. 4 ZHENGQU WAN 2. Boltzmann Distribution Arises from the Principle of Indifference In this section, we will see how Boltzmann distribution arises from the principle of indifference. We will use the ideal gas model for demonstration. In the ideal gas model, N particles move and bounce inside a d-dimensional box Ω. The particles themselves do not collide. The state of the ideal gas is completely specified by the positions X1; :::; XN and momentums P1; :::; PN of all the particles. Once the initial state of the ideal gas is known, the system evolves according to Newtonian mechanics. Just like in the previous game where we are interested in the distribution of wealth among the people, here we are interested in the distribution of kinetic en- ergy among the gas particles. Although the evolution of ideal gas is deterministic, we still treat Xi and Pi as random variables, because the number of gas particles is huge in reality (∼ 1023) and it is impossible to measure the state of the system or to solve the evolution. d Since the position Xi 2 Ω and the momentum Pi 2 R , the state of the ideal gas belongs to the phase space d N (Ω × R ) dkB T At temperature T , each particle particle has average kinetic energy 2 (assume the particle is mono-atomic, kB is the Boltzmann constant) Therefore the total kinetic energy satisfies N 2 X P NdkBT i = 2m 2 i=1 Where m is the mass of each gas particle. Therefore at temperature T , the state of the ideal gas belongs to the constant energy hypersurface N 2 X P NdkBT S = (X; P ): i = = ΩN × L 2m 2 i=1 Where N 2 X P NdkBT L = P : i = 2m 2 i=1 It turns out, by the principle of indifference in statistical mechanics, each state on the energy hypersurface is equally likely to be observed as any other state. Therefore (X; P ) is the uniform random variable on S. We have the following meaningful consequences: (1) Each Xi is a uniform random variable in Ω. (2) The positions of different particles are independent, i.e. fX1; :::; XN g is independent. (3) The positions (X1; :::; Xn) are independent to the momentums (P1; :::; Pn). (4) Each Pi obeys the Boltzmann-Maxwell Distribution as N ! 1. (5) The momentums of different particles are Asymptotically Independent, i.e. the joint distribution of each fP1; :::; Pkg tends to be independent as N ! 1. THE BOLTZMANN DISTRIBUTION 5 (1), (2) and (3) are not too difficult to obtain. I will prove (4) and (5) as they are analogous to the properties emphasized in the firse section: wealth of individual person tends to obey the Boltzmann distribution and tends to be independent. Theorem 2.1 (The 4th statement). The momentum of each particle obeys the Boltzmann-Maxwell distribution as N ! 1, that is, the probability measure on Rd is given by 1 P 2=2m dP r = d exp − dP (2πmkBT ) 2 kBT Where dP is the euclidean measure on the momentum space Rd.
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