Classical phase space Corresponding to a macro state of the system there are thus sets of micro states which belong to the ensemble Phase space and probability density with the probability ½(P ) d¡. ½(P ) is the probability We consider a system of N particles in a d-dimensional density which satis¯es the condition space. Canonical coordinates and momenta Z d¡ ½(P ) = 1: q = (q1; : : : ; qdN ) p = (p1; : : : ; pdN ) The statistical average, or the ensemble expectation value, of a measurable quantity f = f(P ) is determine exactly the microscopic state of the system. Z The phase space is the 2dN-dimensional space f(p; q)g, hfi = d¡ f(P )½(P ): whose every point P = (p; q) corresponds to a possible state of the system. We associate every phase space point with the velocity A trajectory is such a curve in the phase space along ¯eld µ ¶ which the point P (t) as a function of time moves. @H @H V = (q; _ p_) = ; ¡ : Trajectories are determined by the classical equations of @p @q motion The probability current is then V½. The probability dq @H weight of an element ¡ evolves then like i = 0 dt @p Z Z i @ dpi @H ½ d¡ = ¡ V½ ¢ dS: = ¡ ; @t ¡0 @¡0 dt @qi d S where H = H(q1; : : : ; qdN ; p1; : : : ; pdN ; t) G 0 = H(q; p; t) = H(P; t) is the Hamiltonian function of the system. The trajectory is stationary, if H does not depend on Because Z Z time: trajectories starting from the same initial point P V½ ¢ dS = r ¢ (V½) d¡; are identical. @¡0 ¡0 Let F = F (q; p; t) be a property of the system. Now we get in the limit ¡0 ! 0 the continuity equation dF @F @ = + fF; Hg; ½ + r ¢ (V½) = 0: dt @t @t where fF; Gg stands for Poisson brackets According to the equations of motion µ ¶ @H X @F @G @G @F q_ = fF; Gg ´ ¡ : i @p @q @p @q @p i i i i i i @H p_i = ¡ We de¯ne the volume measure of the phase space @qi we have YdN @q_ @p_ dqidpi ¡dN i i d¡ = = h dq1 ¢ ¢ ¢ dqdN dp1 ¢ ¢ ¢ dpdN : + = 0; h @qi @pi i=1 so we end up with the incompressibility condition Here h = 6:62608 ¢ 10¡34Js is the Planck constant. · ¸ X @q_ @p_ Note [dq dp] = Js, so d¡ is dimensionless. r ¢ V = i + i = 0: @q @p Note ¢0¡ = 1 corresponds to the smallest possible i i i volume element of the phase space where a point From the continuity equation we get then representing the system can be localized in accordanceR with the uncertainty principle. The volume ¢¡ = d¡ is @½ 0 = + r ¢ (V½) then roughly equal to the number of quantum states in @t the part of the space under consideration. @½ = + ½r ¢ V + V ¢ r½ The ensemble or statistical set consists, at a given @t moment, of all those phase space points which correspond @½ = + V ¢ r½: to identical macroscopic systems. @t 18 When we employ the convective time derivative We de¯ne the microcanonical ensemble so that its density distribution is d @ = + V ¢ r dt @t 1 µ ¶ ½E(P ) = ±(H(P ) ¡ E): @ X @ @ §E = + q_ +p _ ; @t i @q i @p i i i Every point of the energy surface belongs with the same probability to the microcanonical ensemble. the continuity equation can be written in the form known The microcacnonical ensemble is stationary, i.e. @½E = 0 as the Liouville theorem @t and the expectation values over it temporal constants. d The mixing flow is such an ergodic flow where the points ½(P (t); t) = 0: dt of an energy surface element d¡E disperse in the course of time all over the energy surface. The points in the phase space move like an incompressible If½ ^ (P; t) is an arbitrary non stationary density fluid which carries with it the constant probability E distribution at the moment t = t , then describing the ensemble. 0 1 lim ½^E(P; t) = ±(H(P ) ¡ E) = ½E(P ) Flow in phase space t!1 §E The energy surface ¡E is the manifold determined by the equation and H(q; p) = E: Z lim hfi = lim d¡½ ^E(P; t)f(P ) Since the energy is a constant of motion every phase t!1 t!1 i Z point P (t) moves on a certain energy surface ¡Ei. The expectation value of the energy of the system = d¡ f(P )½E(P ) Z E = hHi = d¡ H½ i.e. the density describing an arbitrary (non equilibrium) state evolves towards a microcanonical ensemble. is also a constant of motion. The volume of the energy surface is Microcanonical ensemble and entropy Z Z If the total energy of a macroscopic system is known exactly its equilibrium state can be described by a §E = d¡E = d¡ ±(H(P ) ¡ E): microcanonical ensemble. The corresponding probability density is The volume of the phase space is 1 Z Z 1 ½E(P ) = ±(H(P ) ¡ E): §E d¡ = dE §E: ¡1 For a convenience we allow the energy to have some "tolerance" and de¯ne Let us consider the element ¢¡E of an energy surface. 1 Non ergodic flow: In the course of time the element ½E;¢E(P ) = θ(E + ¢E ¡ H(P ))θ(H(P ) ¡ E): ZE;¢E ¢¡E traverses only a part of the energy surface ¡E. Here the normalization constant Ergodic flow: Almost all points of the surface ¡E are Z sometimes arbitrarily close to any point in ¢¡E Z = d¡ θ(E + ¢E ¡ H(P ))θ(H(P ) ¡ E) , E;¢E The flow is ergodic if 8f(P ), f(P ) "smooth enough", is the microcanonical state sum or partition function. ¹ ZE;¢E is the number of states contained in the energy f = hfiE slice E < H < E + ¢E (see the volume measure of the holds. Here f¹ is the time average phase space). In the microcanonical ensemble the Z probability is distributed evenly in every allowed part of 1 T the phase space. f¹ = lim dt f(P (t)) T !1 T 0 Entropy and hfiE the energy surface expectation value We de¯ne the Gibbs entropy as Z 1 Z hfiE = d¡E f(P ): §E S = ¡kB d¡ ½(P ) ln ½(P ): 19 Let ¢¡i the volume of the phase space element i and ½i Quantum mechanical ensembles the average probability density in i. The state of the system is, with the probability Systems of identical particles Let H1 be a Hilbert space for one particle. Then the pi = ½i ¢¡i; Hilbert space for N identical particles is in the element i and HN = H1 ­ H1 ­ ¢ ¢ ¢ ­ H1 : X | {z } pi = 1: N copies 1 We chooce the sizes of all elements to be smallest If, for example, jxii 2 H is a position eigenstate the possible, i.e. ¢¡i = 1. Then N-particle state can be written as X X S = ¡kB ¢¡i½i ln ½i = ¡kB ½i¢¡i ln ½i¢¡i jªi = i i ZZ Z X ¢ ¢ ¢ dx1 ¢ ¢ ¢ dxN jx1;:::; xN i Ã(x1;:::; xN ); = ¡kB pi ln pi; i where since ln ¢¡i = 0. If ½ is constant in the range ¢¡ = W we have jx1;:::; xN i = jx1i ­ jx2i ­ ¢ ¢ ¢ ­ jxN i : 1 ½ = ; There are two kinds of particles: W Bosons The wave function is symmetric with respect to so that Z 1 1 the exchange of particles. S = ¡kB ln d¡: W W Fermions The wave function is antisymmetric with We end up with the Boltzmann entropy respect to the exchange of particles. S = kB ln W: Note If the number of translational degrees of freedom is less than 3, e.g. the system is con¯ned to a two Here W is the thermodynamic probability: the number of dimensional plane, the phase gained by the many particle all those states that correspond to the macroscopical wave function under the exchange of particles can be properties of the system. other than §1. Those kind of particles are called anyons. One can show that the entropy is additive, i.e. if the The Hilbert space of a many particle system is not the system is composed of two partial systems 1 and 2 its whole HN but its subspace: entropy is ½ N 1 1 S1+2 = S1 + S2: SH = S(H ­ ¢ ¢ ¢ ­ H ) symm. H = N 1 1 If we require that the entropy has a maximum under the AH = A(H ­ ¢ ¢ ¢ ­ H ) antisymm. condition Z d¡ ½(P ) = 1; Dimension of space and statistics Let us consider two identical particles in an ½ takes the form n-dimensional Euclidean space En. ½(P ) = ½0 8P 2 ¡E: We separate the center of mass and relative coordinates: The maximum principle of the entropy leads thus to the 1 X = (x1 + x2) 2 En microcanonical distribution. 2 x = (x1 ¡ x2) 2 En: Entropy and disorder The maximum of entropy Since the particles are identical we identify the points , x = x ¡ x Microcanonical ensemble 1 2 , ¡x = x2 ¡ x1 Every microscopic state which satis¯es in the space En of the relative motion. Let us call the E < H < E + ¢E; resulting space r(n; 2). The point 0 2 r(n; 2) is the singular point in this space. is present with the same probability, i.e. there is a complete lack of information Two dimensional space , The space r(2; 2) is a circular cone with the vertex Disorder is at maximum. aperture 60±. 20 ² corresponds to a curve connecting points x and ¡x - x in the original space En, i.e. corresponds to the exchange of the particles.