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
Q
Low
Temperature
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
u(X,0)
s(X,0)
T(X,0)
x2
x1
u(X,t)
s(X,t)
T(X,t)
x2
x1
Conservation of energy
TR(X)
δq
TR2
δQ
TR1
δq
δQ
δIk(x)
u(X,t)
nk
δIk
δIk(x+dx)
δQ
TR3
δQ
TR4
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

u 
 Ik   Q
X k
• Elements of area cannot store energy, so heat in and heat
out must be equal
 I k nk   q
δq
TR(X)
nk
δIk
δQ
δIk(x)
u(X,t)
δIk(x+dx)
Internal Variables
• 6 fields of internal variables:
u ( X , t ), s( X , t ), T ( X , t ), I k ( X , t ), Q( X , t ), q( X , t )
• 3 constraints:
– Conservation of energy on the surface
– Conservation of energy in the volume
– Thermodynamic model
• 3 independent internal variables:
I k ( X , t ), Q( X , t ), q( X , t )
δq
TR(X)
nk
δIk
δQ
δIk(x)
u(X,t)
δIk(x+dx)
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:
S  log   log   1 U  T S
U
T
Q

s

• From each thermal reservoir to the volume:
TR
• From each thermal reservoir to the surface:
s 
q
TR
• Integrate over continuum of thermal reservoirs:
 SR  
Q
TR
dV  
q
TR
δq
dA
δQ
TR(X)
Entropy of Conductor
From temperature definition and energy conservation:

u

u

 Ik   Q
 I k nk   q
 SC    sdV   dV
X k
T
A bunch of math:

1

Q
1 
 SC     Q 
 I K  dV  
dV 
 I K dV
T
X k
T
T X k

1 
 1
 1

 IK 

I


I
K 
K

 
T X k
X k  T
X k  T 

 Ik
 1
q


I
dV

n
dA


 X k  T K 
T k
 T dA
 SC  
Q
T
dV  
q
T
dA  
δQ
 1
  I k dV
X k  T 
δq
nk
δIk
δIk(x)
u(X,t)
δIk(x+dx)
Total Entropy
• Total entropy change of the system is the sum of the
entropy of the reservoirs and the pure thermal system
 Stot   S R   SC
• Have equation in terms of variations in our three
independent internal variables
1 1 
1 1 
 1
 Stot       QdV       qdA  
  I k dV
X k  T 
 T TR 
 T TR 
• Fundamental postulate – this total entropy must stay the
same or increase
• Three separate inequalities:
1 1 
   Q  0
 T TR 
1 1 
   q  0
 T TR 

X k
1
  Ik  0
T 
Equilibrium
• No change in the total entropy of the system
1 1 
   Q  0
 T TR 
1 1 
   q  0
 T TR 
 1
  Ik  0
X k  T 
• 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
Non-equilibrium
• Total entropy of the system increases with time
1 1 
1 1 
   Q  0    q  0
 T TR 
 T TR 

X k
1
  Ik  0
T 
• Many ways to fulfill these three inequalities
• Choice depends on material properties and boundary conditions
• Ex. Adiabatic with heat flux linear in temperature gradient:
Q  0 q  0
Ji 
I ( X , t )
T ( X , t )
  (T )
t
X i
 (T )  0
• Ex. Conduction at the surface with heat flux linear in temperature
gradient:
Q  0
q
 K (TR  T )
t
T ( X , t )
J i   (T )
X i
K 0
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
δq>0
T(x,0)=T1<TR
δq=0
Example 1: Rod with thermal reservoir at one
end
TR
δq>0
T(x,0)=T1<TR
δq=0
x
• Thermodynamic model of rod:
– Heat capacity “c” constant within the temperature range
u (T )
c
T
 u  c T
c
 s  T
T
• Kinetic model of rod:
– Heat flux proportional to thermal gradient
– Conductivity “κ” constant within the temperature range
T ( x, t )
J  
x
T ( x, t )
 2T ( x, t )
D
t
x 2
D

c
Example 1: Rod with thermal reservoir at one
end
TR
δq>0
T(x,0)=T1<TR
δq=0
• 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)
Example 1: Rod with thermal reservoir at one
end
TR
δq>0
δq=0
T(x,∞)=T1=TR
 u  c T
U  cV TR  T1 
u
s 
T
TR
S  cV ln
T1
Boundary conditions:
T ( x  0, t )  TR
T ( x  L, t )
0
x
TR
T1
L ~ Dt
Example 2: Rod with thermal reservoirs at
different temperatures at each end
TR1
δq<0
TR1<T(x,0)=T1<TR2
δq>0 T
R2
• 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
Example 2: Rod with thermal reservoirs at
different temperatures at each end
TR1
δq<0
δq>0 T
R2
TR1<T(x,t)<TR2
• System never reaches equilibrium since there is always a
temperature gradient across it
• Steady-state temperature profile is linear
TR2
TR1
 TR1  TR 2

U  cV 
 T1 
2


U ss  0
S ss  0
Stot _ ss
q
q


TR1 TR 2
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
TR
 1 ,  2 ,  3 ,  4 ... s
U1 ,U 2 ,U3 ,U 4 ...U s
isolated system
Boltzmann Factor
• Recall:
 log  1

U
T
• Energy is conserved
U tot  constant
U  T  log 
Utot  U s  U R
Us
log  R U tot  U s   log  R U tot  
TR
 Us 
R U tot  U s    R U tot  exp   
 TR 
Boltzmann Factor
 Us 
R U tot  U s    R U tot  exp   
 TR 
• Number of states of the reservoir as an isolated
system:
R Utot 
• Number of states of reservoir when in contact with
small system in state γs:
R Utot  U s 
• Therefore number of states in reservoir reduced by:
 Us 
exp   
 TR 
Boltzmann Distribution
• Isolated system in equilibrium has equal
probability of being in each state
• Probability of being in a particular state:
Small
system
1
2
3
4
s
Thermal
Reservoir
x
x x x
x
x
x
x
x x
x
x
x x
x
x
x
x x
x
x
x
x
x
x
1*  R U tot  U s 
Ps 
tot
tot   R Utot  U s 
s
Boltzmann Distribution
tot   R Utot  U s 
s
&
 Us 
R U tot  U s    R U tot  exp   
 TR 
 Us 
tot   R U tot   exp   
s
 TR 
• Identify the partition function:
 Us 
Z   exp   
s
 TR 
tot  R Utot  * Z
• Revised expression for probability of state s:
exp 

Ps 
U s
Z

TR 
Configurations
Small
system
Thermal
Reservoir
• Frequently interested
in a macroscopic
1
x x
x x x
x

property
x
2
x
x x
x
x

3
• Subset of states of a
x x x
x
4
x
x
x
system called a
x
x
x
x x
x
s
configuration
• Probability of a
 Us 
configuration (A) is
Z A   exp   
 A
sum of probability of
 TR 
ZA
Ps 
states (s) contained in
Z
 Us 
the configuration
Z   exp   
s
s
 TR 