Heat Conduction and the Boltzmann Distribution

Meredith Silberstein ES.241 Workshop

May 21, 2009

Heat Conduction

• Transfer of thermal energy • Moves from a region of higher

temperature to a region of lower temperature

High Temperature

Low Temperature

QQ

What we can/can’t do with the fundamental postulate

• Can:– Derive framework for heat conduction– Find equilibrium condition– Derive constraints on kinetic laws for

systems not in thermal equilibrium• Cannot:

– Directly find kinetic laws, must be proposed within constraints and verified experimentally (or via microstructural specific based models/theory)

Assumptions• Body consists of a field of material particles• Body is stationary• u, s, and T are a functions of spatial coordinate x

and time t• There are no forms of energy or entropy transfer

other than heat• Energy is conserved• No energy associated with surfaces• A thermodynamic function s(u) is known

x1

x2

u(X,0)s(X,0)T(X,0)

x1

x2

u(X,t)s(X,t)T(X,t)

Conservation of energy

δQ

u(X,t)

TR1

TR(X) TR2

TR3

TR4

nk

δIk

δIk(x+dx)δIk(x)

δq

δq

δQ

δQ

δQ

isolated system

Conservation of energy

• Isolated system: heat must come from either thermal reservoir or neighboring element of body

• Elements of volume will change energy based on the difference between heat in and heat out

• Elements of area cannot store energy, so heat in and heat out must be equal

kk

u I QX

δIkδQ

u(X,t)

TR(X)

nk

δIk(x+dx)δIk(x)

δqk kI n q

Internal Variables

• 6 fields of internal variables:

• 3 constraints:– Conservation of energy on the surface– Conservation of energy in the volume– Thermodynamic model

• 3 independent internal variables:

( , ), ( , ), ( , ), ( , ), ( , ), ( , )ku X t s X t T X t I X t Q X t q X t

( , ), ( , ), ( , )kI X t Q X t q X t

δIkδQ

u(X,t)

TR(X)

nk

δIk(x+dx)δIk(x)

δq

Entropy of reservoirs• Temperature of each reservoir is a constant

(function of location, not of time)• No entropy generated in the reservoir when heat is

transferred• Recall:

• From each thermal reservoir to the volume:

• From each thermal reservoir to the surface:

• Integrate over continuum of thermal reservoirs:

logS log 1

U T

U T S

R

Qs

T

R

qsT

RR R

Q qS dV dA

T T

TR(X)

δQ

δq

Entropy of Conductor

From temperature definition and energy conservation:

C

uS sdV dV

T

kk

u I QX

k kI n q

1 kK k

k

I qI dV n dA dA

X T T T

1 1C K K

k k

QS Q I dV dV I dV

T X T T X

1 1 1K K K

k k k

I I IT X X T X T

δQ

u(X,t)

nk

δIk(x+dx)δIk(x)

δq

δIk

A bunch of math:

1C k

k

Q qS dV dA I dV

T T X T

Total Entropy

• Total entropy change of the system is the sum of the entropy of the reservoirs and the pure thermal system

• Have equation in terms of variations in our three independent internal variables

• Fundamental postulate – this total entropy must stay the same or increase

• Three separate inequalities:

tot R CS S S

1 1 1 1 1tot k

R R k

S QdV qdA I dVT T T T X T

1 10

R

QT T

1 10

R

qT T

10k

k

IX T

Equilibrium

• No change in the total entropy of the system

• The temperature of the body is the same as the temperature of the reservoir

• There is no heat flux through the body– The reservoirs are all at the same temperature

1 10

R

QT T

1 10

R

qT T

10k

k

IX T

Non-equilibrium

• Total entropy of the system increases with time

• Many ways to fulfill these three inequalities• Choice depends on material properties and boundary

conditions• Ex. Adiabatic with heat flux linear in temperature gradient:

• Ex. Conduction at the surface with heat flux linear in temperature gradient:

0Q

1 10

R

QT T

1 10

R

qT T

10k

k

IX T

0q ( , ) ( , )( )i

i

I X t T X tJ T

t X

( ) 0T

0Q ( )R

qK T T

t

( , )

( )ii

T X tJ T

X

0K

Example 1: Rod with thermal reservoir at one end

• Questions:– What is the change in energy and entropy of

the rod when it reaches steady state?– What is the temperature profile at steady-

state?

• Interface between reservoir and end face of rod has infinite conductance

• Rest of surface insulated

TR T(x,0)=T1<TR

δq>0 δq=0

Example 1: Rod with thermal reservoir at one end

• Thermodynamic model of rod:– Heat capacity “c” constant within the temperature range

• Kinetic model of rod:– Heat flux proportional to thermal gradient– Conductivity “κ” constant within the temperature range

( )u Tc

T

TR T(x,0)=T1<TR

δq>0 δq=0

u c T c

s TT

( , )T x tJ

x

2

2

( , ) ( , )T x t T x tD

t x

x

Dc

Example 1: Rod with thermal reservoir at one end

• Heat will flow from reservoir to rod until entire rod is at the reservoir temperature

• Rate of this process is controlled by conductivity of rod

• Change in energy depends on heat capacity (not rate dependent)

TR T(x,0)=T1<TR

δq>0 δq=0

Example 1: Rod with thermal reservoir at one end

1

ln RTS cVT

u c T 1RU cV T T

TR T(x,∞)=T1=TR

δq>0 δq=0

us

T

~L Dt

Boundary conditions:

( , )0

T x L t

x

( 0, ) RT x t T

T1

TR

Example 2: Rod with thermal reservoirs at different temperatures at each end

• Questions:– What is the change in energy and entropy of

the rod when it reaches steady state?– What is the temperature profile at steady-

state?

• Same thermodynamic and kinetic model as rod from first example problem

TR2TR1 TR1<T(x,0)=T1<TR2

δq<0 δq>0

Example 2: Rod with thermal reservoirs at different temperatures at each end

• System never reaches equilibrium since there is always a temperature gradient across it

• Steady-state temperature profile is linear

1 212

R RT TU cV T

TR2TR1 TR1<T(x,t)<TR2

δq<0 δq>0

TR1

TR2

0ssS 0ssU _

1 2tot ss

R R

q qS

T T

Boltzmann Distribution

• Question: What is the probability of a body having a property we are interested in as derived from the fundamental postulate?

• Special case of heat conduction: – Small body in contact with a large reservoir– Thermal contact– No other interactions– Energy exchange without work

• But the body is not an isolated system

Boltzmann Distribution

• No interaction of composite system with rest of environment

• Small system can occupy any set of states of any energy

• System fluctuates among all states while in equilibrium

1 2 3 4, , , ... s TR

isolated system

1 2 3 4, , , ... sU U U U U

Boltzmann Factor

• Recall:

• Energy is conserved

totU constant tot s RU U U

log 1

U T

logU T

log log sR tot s R tot

R

UU U U

T

exp sR tot s R tot

R

UU U U

T

Boltzmann Factor

• Number of states of the reservoir as an isolated system:

• Number of states of reservoir when in contact with small system in state γs:

• Therefore number of states in reservoir reduced by:

R totU

R tot sU U

exp s

R

U

T

exp sR tot s R tot

R

UU U U

T

Boltzmann Distribution

• Isolated system in equilibrium has equal probability of being in each state

• Probability of being in a particular state:

1* R tot ss

tot

U UP

Small system

Thermal Reservoir

x x

x xx

xxx

x xx

x

x

xx

x

xx

x

x

x

x xx

x

s

1234

s

tot R tot sU U

Boltzmann Distribution

exps

s

R

UZ

T

exps

stot R tot

R

UU

T

s

tot R tot sU U

exp sR tot s R tot

R

UU U U

T

&

• Identify the partition function:

• Revised expression for probability of state s:

*tot R totU Z

exp s

Rs

UT

PZ

Configurations

• Frequently interested in a macroscopic property

• Subset of states of a system called a configuration

• Probability of a configuration (A) is sum of probability of states (s) contained in the configuration exp

s

s

R

UZ

T

exps

sA

A R

UZ

T

As

ZP

Z

Small system

Thermal Reservoir

x xx x

xxx

xx x

x

x

x

xx

x

xx

x

x

x

x xx

x

s

1234