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LECTURE 8

THEORY

Closed

system:

energy

E = const

S ~ln[M]

CONSIDER: 1 state of “small part” with & all

states of thermostat with E-. M(E-) = 1•Mth(E-)

St(E-) = k •ln[Mt(E-)] St(E) - •(dSt/dE)|E

Mt(E-) = exp[St(E)/k] • exp[-•(dSt/dE)|E/k]

All-system’s states with E have equal probabilities

For “small part’s” state: depends on e:

COMPARE:

Probability1(1) = Mt(E-1)/SE’M(E’) ~

exp[St(E)/k] • exp[-1• (dSt/dE)|E/k]

and

Probability1(1) ~ exp(-1/kBT) (BOLTZMANN)

One has: (dSt/dE)|E = 1/T

k = kB

______________________________________________________________

-kBT, M Mexp(1) M2.72

P1(1) ~ exp(-1/kBT)

Pj(j) = exp(-j/kBT)/Z(T); j Pj(j) 1

Z(T) = i exp(-i/kBT) partition function

СТАТИСТИЧЕСКАЯ СУММА

Along tangent: S-S(E1)= (E-E1)/T1

i.e.,F = E - T1S= const (= F1 = E1 - T1S1)

stable

Unstable (explodes, V → ): Unstable (fells):

unstable

Separation of potential energy

in classic (non-quantum) mechanics:

P() ~ exp(-/kBT) Classic:=COORD+ KIN

KIN= mv2/2 : does not depend on coordinates

Potential energy COORD: depends only on coordinates

P() ~ exp(-COORD/kBT) • exp(-KIN/kBT)

Z(T) = ZCOORD(T)•ZKIN(T) F(T) = FCOORD(T)+FKIN(T)

========================================================================================================================

Elementary volume: (mv)x = ħ (x)3 =(ħ/|mv|)3

TCOORD=TKIN=T

We may consider further

only potential energy:

EECOORD

MMCOORD

S(E)SCOORD(ECOORD )

F(E)FCOORD , etc.

THERMODYNAMICS

“all-or-none” (or first order)

phase transition

coexistence

(T/T*)(E/kT*)~1

coexistence

& jump

KINETICS

#

_

start

finish

CONSECUTIVE REACTIONS:

TRANSITION TIME SUM OF TIMES

t0… finish = t0#1 finish + t0#2 finish + …

#main

_

_

“trap”: on

“trap”: out

start

start

finish

finish

TRANSITION TIME IS ESSENTIALLY

EQUAL FOR “TRAPS” AT AND OUT OF PATHWAYS OF CONSECUTIVE REACTIONS:

TRANSITION TIME SUM OF TIMES

(or: the longest time)

KINETICS

Mean kinetic energy of a particle:mv2/2 ~ kBT<> = j Pj(j) j v2 = (vX2)+(vY2)+(vZ2)Maxwell:

Friction stops a molecule within picoseconds:

m(dv/dt) = -(3D)v [Stokes law]

D – diameter;

m ~ D3 – mass;

– viscosity

tkinet 10-13 sec (D/nm)2in water

During tkinet the molecule moves somewhere by Lkinet ~ v•tkinet

Then it restores its kinetic energy mv2/2 ~ kBT from thermal kicks of other molecules, and moves in another random side

CHARACTERISTIC DIFFUSION TIME: nanoseconds

Friction stops a molecule within picoseconds:

tkinet 10-13 sec (D/nm)2in water

During tkinet the molecule moves somewhere by Lkinet ~v•tkinet

Then it restores its kinetic energy

mv2/2 ~ kBT from thermal kicks

of other molecules, and moves in

another random side

CHARACTERISTIC DIFFUSION

TIME: nanoseconds

The random walk allows the molecule

to diffuse at distance D (~ its diameter)

within ~(D/L kinet)2steps, i.e., within

tdifft tkinet•(D/Lkinet)2 4•10-10 sec (D/nm)3in water

For “small part”: Pj(j) = exp(-j/kBT)/Z(T);

Z(T) = j exp(-j/kBT)

j Pj(j) = 1

E(T) = <> = j j Pj(j)

if allj =: #STATES = 1/P, i.e.: S(T) = kBln(1/P)

S(T) = kB<ln(#STATES)>= kBj ln[1/Pj(j)]Pj(j)

F(T) = E(T) - TS(T) = -kBT ln[Z(T)]

STATISTICAL MECHANICS

“Small part”: Pj(j,Tth) ~ exp(-j/kBTth);

E(Tth) = j jPj(j,Tth)

S(Tth) = kBj ln[1/Pj(j,Tth)]Pj(j,Tth)

Tsmall_part = dE(Tth)/dS(Tth) = Tth

STATISTICAL

MECHANICS

Separation of potential energy

in classic (non-quantum) mechanics:

P() ~ exp(-/kBT) Classic:=COORD+ KIN

KIN= mv2/2 : does not depend on coordinates

Potential energy COORD: depends only on coordinates

P() ~ exp(-COORD/kBT) • exp(-KIN/kBT)

ZKIN(T) =K exp(-K/kBT): don’t depend on coord.

ZCOORD(T) =Cexp(-C/kBT): depends on coord.

Z(T) = ZCOORD(T)•ZKIN(T) F(T) = FCOORD(T)+FKIN(T)

========================================================================================================================

Elementary volume: (mv)x = ħ (x)3 =(ħ/|mv|)3

P(KIN+COORD) ~ exp(-COORD/kBT)•exp(-KIN/kBT)

P(COORD) = exp(-COORD/kBT) / ZCOORD(T)

ZCOORD(T) =Cexp(-C/kBT): depends ONLY

on coordinates

P(KIN) = exp(-KIN/kBT) / ZKIN(T)

ZKIN(T) =K exp(-K/kBT): don’t depend on coord.

T<0: unstable (explodes)

<KIN> at T<0

due to

P(KIN) ~ exp(-KIN/kBT)

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