Macrostates & Microstates 1

8/19/2019 Macrostates & Microstates 1

1/21

Lecture 4. Macrostates and Microstates (Ch. 2 )

The past three lectures: we have learned about thermal energy, how it is

stored at the microscopic level, and how it can be transferred from onesystem to another. However, the energy conservation law (the first law of

thermodynamics) tells us nothing about the directionality of processes

and cannot explain why so many macroscopic processes are irreversible.

Indeed, according to the st law, all processes that conserve energy are

legitimate, and the reversed!in!time process would also conserve energy.

Thus, we still cannot answer the basic "uestion of thermodynamics: why

does the energy spontaneously flow from the hot ob#ect to the cold ob#ect

and never the other way around$ (in other words, why does the time arrow

exist for macroscopic processes$).

%or the next three lectures, we will address this central problem using theideas of statistical mechanics. &tatistical mechanics is a bridge from

microscopic states to averages. In brief, the answer will be: irreversible

processes are not inevitable, they are just overwhelmingly probable.

This path will bring us to the concept of entropy and the second law of

thermodynamics.


2/21

Microstates and Macrostates

Macrostate: the state of a macro system specified by its macroscopic

parameters. Two systems with the same values of macroscopic parameters

are thermodynamically indistinguishable. ' macrostate tells us nothing about a

state of an individual particle.

%or a given set of constraints (conservation laws), a system can be in many

macrostates.

Microstate: the state of a system specified by describing the "uantum state

of each molecule in the system. %or a classical particle parameters ( x i , y i ,

z i , p xi , pyi , pzi ), for a macro system N parameters.

The statistical approach: to connect the macroscopic observables

(averages) to the probability for a certain microstate to appear along the

systems tra#ectory in configuration space, P (σ

!, σ 2,...,σ ").

The evolution of a system can be represented by a tra#ectory

in the multidimensional (configuration, phase) space of micro!parameters. *ach point in this space represents a microstate.

+uring its evolution, the system will only pass through accessible microstates

the ones that do not violate the conservation laws: e.g., for an isolated system,

the total internal energy must be conserved.

σ !

σ 2

σ i


3/21

#he $hase %pace vs. the %pace o& Macroparameters

V

T

P

σ !

σ 2

σ ithe surface

defined by ane"uation of

states

some macrostate

σ !

σ 2

σ i

σ !

σ 2

σ i

σ !

σ 2

σ i

numerous microstates

in a multi!dimensional

configuration (phase)

space that correspond

the same macrostate

etc., etc., etc. ...


4/21

'amples #wo*+imensional Con&iguration %pace

motion of a particle in a

one!dimensional box

-L L

-L L x

p x

-p x

-acrostates are characteri/ed by a

single parameter: the 0inetic energy K

K

*ach macrostate corresponds to a continuum of

microstates, which are characteri/ed by specifying the

position and momentum

K=K

'nother example: one!dimensional

harmonic oscillator

x

p x

K + U =const

x

U (r )


5/21

#he -undamental ssumption o& %tatistical Mechanics

The ergodic hypothesis an isolated system in an

e"uilibrium state, evolving in time, will pass through all the

accessible microstates at the same recurrence rate, i.e.all accessible microstates are e!"ally probable.

The average over long times will e"ual the average over the ensemble of all e"ui!

energetic microstates: if we ta0e a snapshot of a system with N microstates, we will find

the system in any of these microstates with the sa#e probability.

1robability for a

stationary system

many identical measurements

on a single system

a single measurement on

many copies of the system

The ensemble of all e"ui!energetic states

⇒ a #irocanonical ense#ble.

2ote that the assumption that a system is isolated is important. If a system is co"pled to

a heat reservoir and is able to exchange energy, in order to replace the systems

tra#ectory by an ensemble, we must determine the relative occurrence of states with

di$$erent energies. %or example, an ensemble whose states recurrence rate is given by

their 3olt/mann factor (e-E/kBT ) is called a canonical ensemble.

σ !

σ 2

σ i

microstates which correspond

to the same energy


6/21

$robability o& a Macrostate, Multiplicity

( )

smicrostateaccessibleallof

macrostategivenatocorrespondthatsmicrostateof

macrostateparticular aof y1robabilit

#

#Ω=

=

The probability of a certain macrostate is determined by how many

microstates correspond to this macrostate the multiplicity o& a given

macrostate .

This approach will help us to understand why some of the macrostates are

more probable than the other, and, eventually, by considering the interacting

systems, we will understand irreversibility of processes in macroscopic

systems.

smicrostateaccessibleallof 4

ensembleicalmicrocanonaof microstateparticular aof y1robabilit

=

=


7/21

$robability

%"ltiplication r"le for independent events: P (i and j ) 5 P (i ) x P ( j )

1robability theory is nothing but common sense

reduced to calculations 6aplace (78)

&xa#ple: 9hat is the probability of the same face appearing on two successive

throws of a dice$

The probability of any specific combination, e.g., (,): ;x;5,>) =...= P (;,;) 5 ;x


8/21

#wo model systems with &ied positions o& particles

and discrete energy levels

! the models are attractive because they can be described in terms ofdiscrete microstates which can be easily counted (for a continuum of

microstates, as in the example with a freely moving particle, we still need

to learn how to do this). This si#pli$ies calc"lation o$ . @n the other

hand, the results will be applicable to many other, more complicated

models.

+espite the simplicity of the models, they describe a number ofexperimental systems in a surprisingly precise manner.

! two!state paramagnet ↑↓↓↑↓↑↑↓↓↓↑↓↑....

(limited energy spectrum)

! the *instein model of a solid

(unlimited energy spectrum)


9/21

#he #wo*%tate $aramagnet

The energy of a macrostate:

↑↓ += N N N

B

N ↑

! the number of up spins

N ↓

! the number of down spins

! a system of non!interacting magnetic dipoles in an external magnetic field (, each dipole

can have only two possible orientations along the field, either parallel or any!parallel to this

axis (e.g., a particle with spin A ). 2o "uadratic degrees of freedom (unli0e in an ideal gas,

where the 0inetic energies of molecules are unlimited), the energy spectrum of the particlesis con$ined )ithin a $inite interval o$ & (#ust two allowed energy levels).

µ ! the magnetic moment of an individual dipole (spin)

&

& ! / * µ(

& 2 / 0 µ(

an arbitrary choiceof /ero energy

! µB for µ parallel to (,

=µB for µ anti!parallel to (

The total magnetic moment:(a macroscopic observable)

The energy of a single dipole in the

external magnetic field: Bii ⋅−= µ ε

' particular #icrostate (↑↓↓↑↓↑↑↓↓↓↑↓↑....)

is specified if the directions of all spins are

specified. ' #acrostate is specified by the total

4 of dipoles that point up, N ↑ (the 4 of dipoles

that point down, N ↓ 5 N - N ↑ ).

( ) ( )↑↑↓ −=−=⋅−= N N B N N B B M U 2 µ µ

( ) [ ] ( ) N N N N N N N M −=−==−= ↑↑↓↓↑ 2 µ µ


10/21

'ample

Bonsider two spins. There are four possible configurations of microstates:

% / 2µ

* 2µ

In /ero field, all these microstates have the same energy (degeneracy). 2ote

that the two microstates with % 5C have the same energy even when (≠C:

they belong to the same macrostate, which has multiplicity 5>. Themacrostates can be classified by their moment % and multiplicity

% / 2µ

* 2µ

/ ! 2 !

%or three spins:

% / 1µ µ µ µ *µ *µ *µ *1µ

% / 1µ

µ

*µ

*1µ

/ ! 1 1 !macrostates:


11/21

#he Multiplicity o& #wo*%tate $aramagnet

*ach of the microstates is characteri/ed by N numbers, the number of

e"ually probable microstates 2N , the probability to be in a particular

microstate !2N .

n 3≡

n &actorial / !2....n

3≡

! (exactly one way toarrange /ero ob#ects)

)!(!

!

!!

!),(

↑↑↓↑↑ −

==Ω N N N

N

N N

N N N

%or a two!state paramagnet in /ero field, the energy of all macrostates is

the same (C). ' macrostate is specified by (N , N ↑). Its multiplicity ! the

number of ways of choosing N ↑ ob#ects out of N :

1)0,( =Ω N N N =Ω )1,( ( )2

1)2,( −×=Ω N N N ( ) ( )23

21)3,(× −×−×=Ω

N N N N

( ) ( )[ ]( )!1!

!

123...

1...1),(

−=

××××−−××−×=Ω

nn

N

n

n N N N n N

The multiplicity of a

macrostate of a two!state

paramagnet with (N , N ↑):


12/21

Math re5uired to bridge the gap between ! and !21

Typically, N is h"ge for macroscopic systems, and the multiplicity is

unmanageably large for an *instein solid with C>


13/21

%tirling6s pproimation &or N 3 (N 77!)

N e

N N e N N

N

N N π π 22!

=≈ −

-ultiplicity depends on N F, and we need an approximation for ln(N F):

N N N N −≈ ln!ln

Bhec0:

( ) [ ] N N N x x x x x N N

−≈−=≈++++= ∫ lnlndlnlnN···ln3ln2ln1lnN! 11

-ore accurately:

because ln N ?? N for large N

( ) ( ) ( ) N N N N N N N N −≈++−= ln2ln2

1ln

2

1ln!ln π

N

e

N N

≈!or


14/21

#he $robability o& Macrostates o& a #wo*%tate $M ((/)

(http:stat!www.ber0eley.eduGstar0ava3inHist.htm4controls)

! as the system becomes larger, the

P (N,N ↑) graph becomes moresharply pea0ed:

N 5 ⇒ Ω(,N ↑) 5, >25>, P (,N ↑)5C.

N

N N

N N

N N N N N N P

2

),(

),(

),(

#

),(),( ↑

↑

↑↑↑

Ω=

Ω

Ω=

Ω=

allsmicrostateallof

( ) ( ) ( ) ( )

( )( ) N N N N

N

N N N N N N N

N N

N

N N N

N

e N N e N

e N

N N N

N N N P

2

22!!

!),(

↑↑

↑↑↑↑

−↑↑

−−−↑

−↑

−

↑↑↑

−=

−≈

−=

N ↑

P (, N ↑)

C.

C nC C.JC>< C><

⇒

⇒

N ↑ N ↑

P (, N ↑) P (C>


15/21

Multiplicity and +isorder

In general, we can say that small multiplicity implies order, while large multiplicity

implies disorder. 'n arrangement with large Ω could be achieved by a randomprocess with much greater probability than an arrangement with small Ω.

↑↓↓↑↓↑↑↓↓↓↑↓↑

↑↑↑↑↑↑↑↑↑↑↑

large Ωsmall Ω


16/21

#he 'instein Model o& a %olid

In 8CK, *instein proposed a model that reasonably predicted the thermal

behavior of crystalline solids (a


17/21

#he 'instein Model o& a %olid (cont.)

't high T *ω (the classical limit of large ni):

moleJ/K .24

332

1)2(3

3

1

⋅⇒

=⇒===∑= B B Bi N

i Nk dT

dU T Nk T k N nU ε

soliddU dT ,

89mole

Lead 2.4

:old 2;.4

%ilver 2;.4

Copper 24.;


18/21

#he Multiplicity o& 'instein %olid

( )

!)1(!

!1),(

−−+

=Ω N q

N qq N

Proo$ lets consider N oscillators, schematically represented as follows:

••• ••••• ! ! dots and N *! lines, total ! 0N *! symbols. %or given ! and N , the multiplicity is the number of ways of choosing n of the symbols

to be dots, ".e.d.

The multiplicity of a state of N oscillators (N < atoms) with ! energy "uanta

distributed among these oscillators:

In terms of the total

internal energy U /!

:

( )

( ) !)1(!/

!1/),(

−

−+=Ω

N U

N U U N

ε

ε

Example: The multiplicity of an *instein solid with three atoms and eight units of

energy shared among them( )( ) !)1(!

!1),(

−−+=Ω >,7KC


19/21

Multiplicity o& a Large 'instein %olid ( (T )

( )

( )

( )

( ) ( )[ ] [ ] [ ]

( )

( ) ( ) ( )

( ) ( ) ( ) N N qq N q N q

N N N qqq N q N q N q

aaaa

N q N q N q

N q

N q

N qq N

lnlnln

lnlnln

ln!

!ln!ln!ln!!

!ln

)!1(!

!1ln),(ln

−−++=

+−+−+−++≈−≈

−−+=

+≈

−−+=Ω

ln:&tirling

q

N q

q

N q N q +≈

+=+ ln1ln)ln(

( ) N N

N N

q N

e N

eqeeq N β =

=≈Ω

ln

),(

! = U

=

N ! the total 4 of "uanta in a solid.

= U ( N ) ! the average 4 of "uanta (microstates) available for each molecule

high te#perat"res

( (T , 77!, ! N )

.eneral state#ent for any system with N "uadratic degrees of freedom

(unlimited spectrum), the multiplicity is proportional to U N 2.

*instein solid:(>N degrees

of freedom)

N

N

U N f N

eU N U )(),( =

≈Ω

ε

( ) ( )

N N

q N N N

q

N N q N

N N qqq

N q N qq N

+≈−++=

−−

++=Ω

lnlnln

lnlnln),(ln

2


20/21

Multiplicity o& a Large 'instein %olid ( (T // )

lo) te#perat"res

( (T // , ==!, ! // N )

β

β

N q

e

q

eN q N

=

=Ω ),( (Pr . >.K)


21/21

Concepts o& %tatistical Mechanics

!. #he macrostate is specified by a sufficient number of macroscopicallymeasurable parameters (for an *instein solid N and U ).

2. #he microstate is specified by the "uantum state of each particle in a

system (for an *instein solid 4 of the "uanta of energy for each of N

oscillators)

1. #he multiplicity is the number of microstates in a macrostate. %oreach macrostate, there is an extremely large number of possible

microstates that are macroscopically indistinguishable.

4. #he -undamental ssumption &or an isolated system, all

accessible microstate are e5ually li>ely.

;. #he probability o& a macrostate is proportional to its multiplicity. Thiswill sufficient to explain irreversibility.

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Macrostates & Microstates 1