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Introduction to statistical mechanics mechanics.. The macroscopic and the microscopic states. states. Equilibrium and observation time time.. Equilibrium and molecular motion motion.. Relaxation time time.. Local equilibrium equilibrium.. Phase space p of a classical system system. y . Statistical ensemble ensemble.. Liouville’s theorem. theorem. Density matrix in statistical mechanics and its properties.. properties Liouville’s Li Liouville’sill ’ -Neiman N i equation equation. ti . 1 Introduction to statistical mechanics. F From th the seventeenth t th century t onward d it was realized li d th thatt material systems could often be described by a small number of descriptive parameters that were related to one another in simple lawlike ways. These parameters referred to geometric, geometric dynamical and thermal properties of matter. Typical T i l off the th laws l was the th ideal id l gas law l that th t related l t d product of pressure and volume of a gas to the temperature of the gas gas. 2 Bernoulli (1738) Joule (1851) Krönig (1856) Cl Clausius i (1857) C. Maxwell (1860) L Boltzmann L. B lt (1871) J. Loschmidt (1876) H. Poincaré ((1890)) J. Gibbs (1902) Planck ((1900)) Langevin (1908) Compton (1923) Smoluchowski (1906) Bose (1924) Debye (1912) T Ehrenfest T. Ei t i (1905) Einstein Pauli (1925) Thomas (1927) Dirac (1927) Fermi (1926) Landau (1927) 3 Energy States Unstable: falling or rolling Stable M t t bl : Metastable: Metastable in lowlow-energy perch Figure 5-1. Stability states. Winter (2001) An Introduction to Igneous and Metamorphic Petrolog Prentice Hall. We work with systems which are in equilibrium How do we define equilibrium q ? The system can be in Mechanical, Chemical and d Thermal h l equilibrium ilib i We call all these three together as “Thermodynamic Equilibrium . Equilibrium” Which means : all the energy gy states are equally q y accessible for all the particles H How d do we di distinguish i i hb between Cl Classical i l and d Quantum systems? 5 Mi Microscopic i and d macroscopic i states The main aim of this course is the investigation of general properties of the macroscopic systems with a large number of degrees of dynamically freedom (with N ~ 1020 particles for example). From the mechanical point of view, such systems are very complicated. But in the usual case only a few physical parameters, say temperature, the pressure and d the th density, d it are measured, d by b means off which hi h the th ’’state’’ ’’ t t ’’ off the system is specified. A state defined in this cruder manner is called a macroscopic state or thermodynamic state. On the other hand, from a dynamical point of view, each state of a system can be defined, at least in principle, as precisely as possible by specifying all of the dynamical variables of the system. system Such a state is called a microscopic state. state 6 Properties of individual molecules Position Molecular geometry Intermolecular forces Properties of bulk fluid ( (macroscopic i properties) ti ) Pressure Internal Energy H t Capacity Heat C it Entropy Viscosity What we know Solution S l ti to t S Schrodinger h di equation ti (Eigen-value (Ei l problem) h2 2 i2 U E Wave function i 8 mi Allowed energy levels : E n Using the molecular partition function, we can calculate average values of property at given STATE QUANTUM STATE. Quantum states are changing so rapidly that the observed dynamic properties are actually time average over quantum states. Definition and Features the Thermodynamic y Method Thermodynamics is a macroscopic, phenomenological theoryy of heat. Basic features of the thermodynamic method: • Multi-particle physical systems is described by means of a small number of macroscopically measurable parameters, the thermodynamic parameters: V, P, T, S (volume, pressure, temperature, entropy), and others. Note: macroscopic objects contain ~ 1023…1024 atoms (Avogadro’s number ~ 6x1023mol– 1) . • The connections between thermodynamic parameters are found from the general laws of thermodynamics. • The laws of thermodynamics are regarded as experimental facts. Therefore, thermodynamics is a phenomenological theory. • Thermodynamics is in fact a theory of equilibrium states, i.e. the states with timeindependent (relaxed) V, P, T and S. Term “dynamics” is understood only in the sense “how one thermodynamic parameters varies with a change of another parameter in two successive equilibrium states of the system”. Classification of Thermodynamic Parameters Internal and external parameters: • External parameters can be prescribed by means of external influences on the system by specifying external boundaries and fields. • Internal parameters are determined by the state of the system itself for given values of the external parameters. Note: the same parameter may appear as external in one system, and as internal in another system. Intensive and extensive parameters: p • Intensive parameters are independent of the number of particles in the system, and they serve as general characteristics of the thermal atomic motion (temperature, chemical potential). • Extensive parameters are proportional to the total mass or the number of particles in the system (internal energy, energy entropy) entropy). Note: this classification is invariant with respect to the choice of a system. Internal and External Parameters: Examples A same parameter may appear both as external and internal in various systems: System A T Const System B M P V = Const External parameter: V Internal parameter: P P= Const V External parameter: P, P = Mg/A Internal parameter: V, V = Ah State Vector and State Equation Application of the thermodynamic method implies that the system if found in the state of thermodynamic equilibrium, denoted X, which is defined by time-invariant state parameters, such as volume, temperature p and p pressure: X (V , T , P ) The parameters (V,T,P) (V T P) are macroscopically measurable measurable. One or two of them may be replaced by non-measurable parameters, such internal energy or entropy. Note that only the mean quantity of a state parameter A is time-invariant, see the plot. A mathematical relationship that involve a complete set of measurable parameters (V,T,P) (V T P) is called the thermodynamic state equation f (V , T , P, ) 0 Here, ξ is the vector of system parameters Averaging The physical quantities observed in the macroscopic state are the result of these variables averaging in the warrantable microscopic states. The statistical hypothesis about the microscopic state distribution is required for the correct averaging averaging. To find the right method of averaging is the fundamental principle of the statistical method for investigation of macroscopic systems. The derivation Th d i ti off generall physical h i l lows l f from th experimental the i t l results lt without consideration of the atomic-molecular structure is the main principle of thermodynamic approach. 13 Averaging Method Probability of observing particular quantum state i ni Pi n i i E Ensemble bl average off a d dynamic i property t E Ei Pi i Time average and ensemble average U lim Ei ti lim Ei Pi n i Thermodynamics and Statistical Mechanics Probabilities 15 Pair of Dice For one die, the probability of any face coming i up is i the th same, 1/6. 1/6 Therefore, Th f it is equally probable that any number from one to six will come up. For two dice, what is the probability that the total will come up 2, 3, 4, etc up to 12? 16 Probability To calculate the probability of a particular ti l outcome, t countt th the number b off all possible results. Then count the number that give the desired outcome. The probability of the desired outcome is equal to the number that gives the desired outcome divided by the total number of outcomes. Hence, 1/6 for one di die. 17 Pair of Dice List all dice. dice Total 2 3 4 5 6 possible outcomes (36) for a pair of Combinations How Many 1+1 1 1+2,, 2+1 2 1+3, 3+1, 2+2 3 1+4 4+1, 4+1 2+3, 2+3 3+2 1+4, 4 1+5, 5+1, 2+4, 4+2, 3+3 5 18 Pair of Dice Total 7 8 9 10 11 12 Combinations How Many 1+6 6+1, 1+6, 6+1 2+5, 2+5 5+2, 5+2 3+4, 3+4 4+3 2+6, 6+2, 3+5, 5+3, 4+4 3+6 6+3, 3+6, 6+3 4+5, 4+5 5+4 4+6, 6+4, 5+5 5+6, 6+5 6+6 6 5 4 3 2 1 Sum = 36 19 Probabilities for Two Dice Total 2 1 Prob. 36 % 2.8 28 3 2 36 56 5.6 4 3 36 83 8.3 5 4 36 11 6 5 36 14 7 6 36 17 8 5 36 14 9 10 11 12 4 3 2 1 36 36 36 36 11 8.3 8 3 5.6 5 6 2.8 28 20 Probabilities for Two Dice Probab bility Dice 0.18 0 16 0.16 0.14 0.12 01 0.1 0.08 0.06 0.04 0 02 0.02 0 2 3 4 5 6 7 8 Number 9 10 11 12 21 Microstates and Macrostates Each possible outcome is called a “microstate”. The combination of all microstates that give the same number of spots is called a “macrostate”. The macrostate that contains the most microstates is the most probable to occur. 22 Combining g Probabilities If a given i outcome t can b be reached h d in i two t (or ( more) mutually exclusive ways whose probabilities b biliti are pA and d pB, then th the th probability b bilit of that outcome is: pA + pB. This is the p probabilityy of having g either A or B. 23 Combining Probabilities If a given outcome represents the combination of two independent events, events whose individual probabilities are pA and pB, then the probability of that outcome is: pA × pB. This is the probability of having both A and B. 24 Example Paint two faces of a die red. When the di is die i thrown, th what h t is i the th probability b bilit off a red face coming up? 1 1 1 p 6 6 3 25 Another Example Throw two normal dice. What is the probability b bilit off two t sixes i coming i up?? 1 1 1 p ( 2) 6 6 36 26 Complications p is the p probabilityy of success. (1/6 ( / for one die) q is the probability of failure failure. (5/6 for one die) p + q = 1, or q=1–p When two dice are thrown thrown, what is the probability of getting only one six? 27 Complications Probabilityy of the six on the first die and not the second is: 1 5 5 pq 6 6 36 Probability of the six on the second die and not the first is the same same, so: 10 5 p (1) 2 pq 36 18 28 Simplification p Probability of no sixes coming up is: 5 5 25 p (0) qq 6 6 36 The sum of all three probabilities is: p(2) + p(1) + p(0) = 1 29 Simplification p(2) + p(1) + p(0) = 1 p² + 2pq + q² =1 (p + q))² = 1 The exponent is the number of dice (or tries). Is this general? 30 Three Dice (p + q)³ = 1 p³ + 3p²q + 3pq² + q³ = 1 p(3) + p(2) + p(1) + p(0) = 1 It works! It must be general! (p + q)N = 1 31 Binomial Distribution Probability of n successes in N attempts (p + q)N = 1 N! n N n P ( n) p q n!( N n)! where, q = 1 – p. 32 Thermodynamic Probability The term with all the factorials in the previous equation is the number of i t t th ill lead l d to t the th particular ti l microstates thatt will macrostate. It is called the “thermodynamic probability”, wn. N! wn n! ( N n ))! 33 Microstates The total number of microstates is: w wn True p probability P(n) For a very large number of particles i l w max 34 Mean of Binomial Distribution n P ( n) n n where N! n N n P ( n) p q n!( N n)! Notice : p P (n) P (n)n p 35 Mean of Binomial Distribution n P ( n) n p P ( n) p n n N n p P ( n) p ( p q ) p n p n pN ( p q ) N 1 pN (1) N 1 n pN 36 Standard Deviation () n n 2 n n P(n)n n 2 2 2 n n n 2 n 2n n n n 2n n n 2 2 n n 2 2 2 2 2 37 Standard Deviation 2 n P ( n) n p P ( n) n p n 2 N N 1 n p p ( p q ) p pN ( p q ) p p p 2 n pN ( p q ) 2 2 N 1 p pN ( N 1)( p q ) N 2 n 2 pN 1 pN p pN q pN 38 Standard Deviation n n 2 2 2 2 pN q pN ( pN ) 2 Npq ( pN ) ( pN ) Npq Npq pq 2 2 2 39 For a Binomial Distribution n pN Npq q Np p n 40 Coins Toss 6 coins. Probabilityy of n heads: n 6 n N! 6! 1 1 n N n P ( n) p q n!( N n)! n!(6 n)! 2 2 6! 1 P ( n) n!(6 n)! 2 6 41 For Six Coins Binomial Distribution 0.35 03 0.3 Probab bilty 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 Successes 4 5 6 42 For 100 Coins Binomial Distribution 0.09 0.08 0.06 0.05 0.04 0.03 0.02 0.01 96 90 84 78 72 66 60 54 48 42 36 30 24 18 12 6 0 0 Proba abilty 0.07 Successes 43 For 1000 Coins Binomial Distribution 0.03 0.02 0.015 0.01 0.005 960 900 840 780 720 660 600 540 480 420 360 300 240 180 120 60 0 0 Probabilty 0 025 0.025 Successes 44 Multiple p Outcomes N! N! w N1! N 2 ! N 3! N i ! N i N i 45 Stirling’s Approximation For large N : ln N ! N ln N N N! ln N ! ln N i ! ln N ! ln N i ! ln w ln i Ni! ln w N ln N N ( N i ln N i ) N i i i ln w N ln N ( N i ln N i ) i 46 Number Expected T Toss 6 coins i N times. ti Probability P b bilit off n heads: h d n 6 n N! 6! 1 1 n N n P ( n) p q n!( N n)! n!(6 n)! 2 2 6 6! 1 P ( n) n!(6 n)! 2 N b off ti d is i expected t d is: i Number times n h heads n = N P(n) 47 Zero Low of Thermodynamics One of the main significant points in thermodynamics (some times they call it the zero low of thermodynamics) is the conclusion that every enclosure (isolated from others) system in time come into the equilibrium state where all the physical parameters characterizing the system are not changing in time. The process of equilibrium setting is called the relaxation p process off the system y and the time off this p process is the relaxation time. Equilibrium means that the separate parts of the system (subsystems) are also in the state of internal equilibrium (if one will isolate them nothing will happen with them). The are also in equilibrium with each other- no exchange g byy energy gy and p particles between them. 48 Local equilibrium Local equilibrium means that the system is consist from the subsystems subsystems, that by themselves are in the state of internal equilibrium but there is no any equilibrium q between the subsystems. y The number of macroscopic parameters is increasing with digression of the system from the total equilibrium 49 Classical phase system Let (q1, q2 ..... qs) be the generalized coordinates of a system with i degrees of freedom and (p1 p2..... ps) their conjugate moment. t A microscopic i i state t t off the th system t i defined is d fi d by b specifying the values of (q1, q2 ..... qs, p1 p2..... ps). The 2s-dimensional space constructed from these 2s variables as the coordinates in the phase space of the system. Each point in the phase space (phase point) corresponds to a microscopic state. Therefore the microscopic states in classical statistical mechanics make a continuous set of ppoints in pphase space. 50 Ph Phase Space S pN t2 2 t1 1 Phase space p1 , p 2 , p 3 ,..., p N , r1 , r1 , r3 ,..., rN rN Phase Orbit If the Hamiltonian of the system is denoted by H(q,p), the motion of phase point can be along the phase orbit and is determined by the canonical equation of motion H qi pi H pi qi P (i=1 2 s) (i=1,2....s) H ( q, p ) E (1 1) (1.1) (1.2) Phase Orbit Constant energy surface H(q,p)=E Therefore the phase orbit must lie on a surface of constant energy (ergodic ergodic surface surface). 52 - space and -space space Let us define - space as phase space of one particle (atom or molecule). molecule) The macrosystem phase space (-space) space is equal to the sum of - spaces spaces. The set of possible microstates can be presented by continues set of phase points. Every point can move by itself along it’s own phase orbit. The overall picture of this movement possesses certain interesting features, which are best appreciated in terms of what we call a density function (q,p;t). y that at anyy time t, the number of This function is defined in such a way representative points in the ’volume element’ (d3Nq d3Np) around the point (q,p) of the phase space is given by the product (q,p;t) d3Nq d3Np. Clearly, the density function (q,p;t) symbolizes the manner in which the members of the ensemble are distributed over various possible microstates at various instants of time. 53 Function of Statistical Distribution Let us suppose that the probability of system detection in the volume ddpdqdp1.... dps dq1..... dqs near point (p,q) equal dw (p,q)= (q,p)d. The function of statistical distribution (density function) of the system over microstates in the case of nonequilibrium systems is also depends on time. The statistical average of a given dynamical physical quantity f(p,q) is equal q f f ( p, q ) (q, p; t )d 3 N qd 3 N p 3N 3N ( q , p ; t ) d qd p (1.3) The right g ’’phase p pportrait’’ off the system y can be described byy the set off points distributed in phase space with the density . This number can be considered as the description of great (number of points) number of systems each of which has the same structure as the system under observation copies of such system at particular time, which are by themselves existing in admissible microstates 54 Statistical Ensemble The number Th b off macroscopically i ll identical id ti l systems t di t ib t d along distributed l admissible microstates with density defined as statistical ensemble. ensemble A statistical ensembles are defined and named by the distribution function which characterizes it. The statistical average value have the same meaning as the ensemble average value. An ensemble A bl is i said id to be b stationary i if does d not depend d d explicitly li i l on time, i.e. at all times 0 t (1 4) (1.4) Clearly, for such an ensemble the average value <f> of any physical Clearly quantity f(p,q) will be independent of time. time Naturally, then, a stationary ensemble qualifies to represent a system in equilibrium. To determine the circumstances i under d which hi h Eq. E (1.4) (1 4) can hold, h ld we have h to make k a rather h study of the movement of the representative points in the phase space. 55 Lioville’s Lioville s theorem and its consequences Consider C id an arbitrary bit " l "volume" " in i the th relevant l t region i off the th phase h space and let the "surface” enclosing this volume increases with time is given by t d (1.5) where h d(d3Nq d3Np) p). ). On O the th other th hand, h d the th nett rate t att which hi h the th representative points ‘’flow’’ out of the volume (across the bounding surface ) is given by ρ(ν n )dσ (1.6) σ here v is the vector of the representative points in the region of the surface element d, while n̂ is the (outward) unit vector normal to this element. l B the By h divergence di theorem, h (1 6) can be (1.6) b written i as 56 div ( v )d ((1.7)) where the operation of divergence means the following: di ( v ) ( qi ) div ( pi ) pi i 1 qi 3N (1.8) In view of the fact that there are no "sources" or "sinks" in the phase p space and hence the total number of representative points must be conserved, we have , by (1.5) and (1.7) div( v )d t d or t div( v )d 0 t d (1.9) (1.10) 57 The necessary and sufficient condition that the volume integral (1.10) vanish for arbitrary y volumes is that the integrated g must vanish everywhere in the relevant region of the phase space. Thus, we must have div ( v ) 0 t (1.11) which is the equation of continuity for the swarm of the representative points. This equation means that ensemble of the phase points moving with time as a flow of liquid without sources or sinks. Combining (1.8) and (1.11), we obtain div( v ) ( qi ) ( pi ) pi i 1 qi 3N 58 q p i qi pi 0 t i 1 qi pi pi i 1 qi 3N 3N (1 12) (1.12) The last group of terms vanishes identically because the equation of motion, we have for all i, qi H ( qi , pi ) H ( qi , pi ) p qi qi pi qi pi pi 2 2 (1.13) From (1.12), taking into account (1.13) we can easily get the Liouville equation ρ 3N ρ ρ ρ qi pi ρ,H 0 t i 1 qi pi t (1.14) where {,H} the Poisson bracket. 59 Further, since (qi,pi;t), t), the remaining terms in (1.12) may be combined to give the «total» time derivative of . Thus we finally have d ,H 0 dt t (1.15) Equation (1.15) embodies the so-called Liouville’s theorem. According to this theorem (q0,p0;t0)=(q,p;t) or for the equilibrium system y (qq0,pp0)= (q,p), q p that means the distribution function is the integral g of motion. One can formulate the Liouville’s theorem as a principle of phase volume maintenance. p t t=0 0 q 60 Density y matrix in statistical mechanics The microstates in quantum theory will be characterized by a H (common) Hamiltonian, which may be denoted by the operator. At time t the physical state of the various systems will be characterized by the correspondent wave functions (ri,t), where the ri, denote the position coordinates relevant to the system under study. study Let k(ri,t) t), denote the (normalized) wave function characterizing the physical state in which the k-th system of the ensemble happens to be at time t ; naturally, k=1,2....N. The time variation of the function k(t) will be determined by the Schredinger equation 61 H k ( t ) i k ( t ) (1.16) Introducing a complete set of orthonormal functions n, the wave functions k(t) may be written as k (t ) ank (t ) n (1 17) (1.17) ank (t ) n k (t ) d (1.18) n here, n* denotes h d the h complex l conjugate j off n while hil d denotes d the h volume element of the coordinate space of the given system. Obviously enough, enough the physical state of the k-th system can be described equally well in terms of the coefficients . The time variation of these coefficients will be given by 62 iank (t ) i n* k (t ) d n*H k (t ) d = * n H m k am (t ) m d = Hnmamk (t ) (1.19) m where H nm n*H m d (1.20) k The physical significance of the coefficients a n (t ) is evident from eqn. ((1.17). ) Theyy are the p probability y amplitudes p for the k-th system y of the ensemble to be in the respective states n; to be practical the number 2 k a n ( t ) represents the probability that a measurement at time t finds the k-th system of the ensemble to be in particular state n. Clearly, we must have 63 2 k a n (t ) 1 (for all k) (1.21) n We now no introduce int od e the density densit operator ope ato (t ) as defined by b the matrix mat i elements (density matrix) 1 mn (t ) N a N k 1 k m (t )ank * (t ) (1.22) clearly, the matrix element mn(t) is the ensemble average of the quantity am(t)an*(t) which, which as a rule rule, varies from member to member in the ensemble. In particular, the diagonal element mn(t) is the ensemble average of the probability, a nk ( t ) 2 the latter itself being a (quantummechanical) average. 64 Equation of Motion for the Density Matrix mn(t) Thus, we are concerned here with a double averaging process - once due to the probabilistic aspect of the wave functions and again due to the statistical aspect of the ensemble!! The quantity mn(t) now represents the probability that a system, chosen at random from the ensemble,, at time t, is found to be in the 2 state n. In view of (1.21) and (1.22) we have ank (t ) 1 n nn 1 n mn (t ) 1 N a N k 1 k m (t )ank * (t ) (1.23) Let us determine the equation of motion for the density matrix mn(t (t). 65 1 i mn ( t ) N i a mk ( t ) a nk * ( t ) a mk ( t ) a nk * ( t ) N k 1 k* k k * k* H ml a l ( t ) a n ( t ) a m ( t ) H nl a l ( t ) k 1 l l = H ml ln ( t ) ml ( t ) H ln 1 = N N l = ( H H ) mn (1.24) Here, use has been made of the fact that, in view of the Hermitian character of the operator, ĤH*nl=Hln. Using the commutator notation, Eq.(1.24) may be written as i H, 0 t (1.25) 66 This equation Liouville-Neiman is the quantum-mechanical analogue of the classical equation Liouville. Some properties of density matrix: •Density Density operator is Hermitian, += - •The density operator is normalized •Diagonal elements of density matrix are non negative •Represent the probability of physical values nn 1 0 n 67