麻省理工大学：《生物工程》教学讲义（英文版）tutorial stat

STATISTICAL MECHANICS BEH.410 Tutorial Maxine jonas February 14, 2003

STATISTICAL MECHANICS BEH.410 Tutorial Maxine Jonas February 14, 2003

Why Statistical Mechanics Understand predict the physical properties of macroscopic systems from the properties of their constituents Deterministic approach ma=F need of 6N coordinates at t r: and u but typically n= moles (1023)! “ Ensemble” rather than microscopic detail and its surroundings microcanonical, canonical, grand canonical

Why Statistical Mechanics? Understand & predict the physical properties of macroscopic systems from the properties of their constituents Deterministic approach - n e e d o f 6N coordinates at t 0: ri and vi - b u t t y p i c a l l y N ≡ moles (1023) ! “Ensemble” rather than microscopic detail … and its surroundings ¾microcanonical, canonical, grand canonical

What with Statistical mechanics? Averages, distributions, deviation estimates of microstates: specification of the complete set of positions and momenta at any given time ( points on the constant energy hypersurface for Hamiltonian dynamics Ensemble average ergodic hypothesis A=(numbe va(e)=iIF a(x(t))dt T time, will visit all possible microscopic states availableto ant of A system that is ergodic is one which, given an infinite amo

What With Statistical Mechanics? Averages, distributions, deviation estimates… … of microstates: specification of the complete set of positions and momenta at any given time (points on the constant energy hypersurface for Hamiltonian dynamics) Ensemble average & ergodic hypothesis: A system that is ergodic is one which, given an infinite amount of time, will visit all possible microscopic states available to it

The first Law- Work Work, heat energy basic concepts Energy of a system = capacity to do work At the molecular level, difference in the surroundings Energy transfer that makes use of Heat chaotic molecular motion Work .organized molecular motion △U=q+w state function -independent of how state was reached

The First Law – Work Work, heat & energy = basic concepts Energy of a system = capacity to do work ¾ At the molecular level, difference in the surroundings state function – independent of how state was reached Energy transfer that makes use of… Heat … chaotic molecular motion Work … organized molecular motion

Second law- Gibbs Spontaneous processes increase the overall"disorder of the universe Reasoning through an example microstates to achieve macrostate Gibbs postulate: for an isolated system, all microstates compatible with the given constraints of the macrostate Chere E,v and w are equally likely to occur Here 2 ways to distribute n molecules into 2 bulbs

Second Law – Gibbs ƒ Spontaneous processes increase the overall “disorder” of the universe ™ Reasoning through an example - microstates to achieve macrostate Gibbs postulate: for an isolated system, all microstates compatible with the given constraints of the macrostate (here E, V and N) are equally likely to occur - Here 2N ways to distribute N molecules into 2 bulbs

Second law- probability Number of (indistinguishable) ways of placing L of the molecules in the left bulb N W L! ( N-L Probability WL/2 maximum if L=N/2 √ With n=1023,p(L=R±1010)=10434 possible but extremely unlikely

Second Law - Probability L ƒ Number of (indistinguishable) ways of placing L of the N molecules in the left bulb: ƒ Probability WL / 2N maximum if L = N / 2 9 With N = 1023, p ( L = R ± 10-10 ) = 10-434 possible but extremely unlikely

Second Law-Entropy Boltzmann’ s constant s=kInw k=138x1023JK1 ∑ pi n p ● Principle of fair Apportionment N Multiplicity of outcomes n1!,!n, N N n(n2(n n n2n, P! P2P

Second Law - Entropy Boltzmann’s constant k = 1.38 x 10-23 J.K-1 Principle of Fair Apportionment Multiplicity of outcomes

Second Law-Entropy The absolute entropy s=kInw Is never negative ∑PlnP S≥0 S max at equilibrium pi InW==>nIn P2 0 0.69 t=1 Order l/21/2 N k Nk N hnW=∑P n e s w 133 139 Flat distribution= high s 31414 “间邮

Second Law - Entropy The absolute entropy is never negative S ≥ 0 S max at equilibrium 0 0rder 1.33 1.39 0.69 Flat distribution ≡ high S pi 1 0 0 0 0 0 n e s w 1/2 1/2 1/3 1/3 1/6 1/6 1/4 1/4 1/4 1/4

The boltzmann distribution law Maximum entropy principle constraints E B3容冷S=∑P ∑PE N ∑ exponential distribution E kT ∑ Partition function @=Expl

The Boltzmann Distribution Law ƒ Maximum entropy principle + constraints E 3 E 2 E 1 ⇒ exponential distribution Partition function

The Boltzmann Distribution Law(2) E More particles have low energy kT more arrangements that way hight.Q= connection between microscopic models macroscopic thermodynamic properties U=kT(OIng and S=kIng+hp/ aIn Q OT medium t- Q= number of states effectively accessible to system Q=∑ep E teEn/kr -Et/kr kT I+e-2/kT E T→+∞0→→0→0>1+1+1+..+1=t low T RT

The Boltzmann Distribution Law (2) ƒ Q ≡ number of states effectively accessible to system E low T E high T E medium T ƒ Q ≡ connection between microscopic models & macroscopic thermodynamic properties and ƒ More particles have low energy: more arrangements that way