# Heat Conduction and the Boltzmann Distribution

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```					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

Q
High                  Low
Temperature           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)                        u(X,t)
x2
s(X,0)             x2
s(X,t)
T(X,0)                        T(X,t)
x1                             x1
Conservation of energy

TR(X)                         TR2
δq
δQ
nk
TR1   δq
δQ                             δIk

δIk(x)                    δIk(x+dx)               TR3
u(X,t)                        δQ
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               TR(X)         δq

nk
δQ                     δIk

δIk(x)                 δIk(x+dx)
u(X,t)
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 )          TR(X)         δq

nk
δQ                      δIk

δIk(x)                 δIk(x+dx)
u(X,t)
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
• From each thermal reservoir to the volume:         s 
TR
q
• From each thermal reservoir to the surface:          s 
TR
• Integrate over continuum of thermal reservoirs:
Q            q                δq
 SR          dV          dA
TR            TR
δQ
TR(X)
Entropy of Conductor
From temperature definition and energy conservation:
u        
 SC    sdV   dV u        Ik   Q  I k nk   q
T        X k
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           IK    IK       
T X k        X k  T              X k  T 

 1                I            q
 I K  dV   k nk dA    dA
 X k  T 
                 T            T

δQ   δq
Q            q              1
 SC          dV          dA            I k dV                                  nk
T             T             X k  T 
δIk

δIk(x)             δIk(x+dx)
u(X,t)
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
 T TR          T TR         X k   T 
• Fundamental postulate – this total entropy must stay the
same or increase
• Three separate inequalities:
1 1            1 1                       1
   Q  0        q  0                   Ik  0
 T TR           T TR               X k   T 
Equilibrium

• No change in the total entropy of the system
1 1             1 1                   1
   Q  0         q  0               Ik  0
 T TR            T TR           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                                   1
   Q  0    q  0                               Ik  0
 T TR       T TR                           X k   T 
• Many ways to fulfill these three inequalities
• Choice depends on material properties and boundary conditions
I ( X , t )           T ( X , t )
Q  0 q  0           Ji 
t
  (T )
X i
 (T )  0
• Ex. Conduction at the surface with heat flux linear in temperature
q                                        T ( X , t )    K 0
Q  0           K (TR  T )            J i   (T )
t                                          X i
Example 1: Rod with thermal reservoir at one
end

• Questions:
– What is the change in energy and entropy of the rod
– What is the temperature profile at steady-state?
• Interface between reservoir and end face of rod
has infinite conductance
• Rest of surface insulated

δq>0                                  δq=0
TR                   T(x,0)=T1<TR
Example 1: Rod with thermal reservoir at one
end
δq>0                                                 δq=0
TR                           T(x,0)=T1<TR
x
• Thermodynamic model of rod:
– Heat capacity “c” constant within the temperature range
u (T )                                          c
c
T
 u  c T               s  T
T
• Kinetic model of rod:
– Heat flux proportional to thermal gradient
– Conductivity “κ” constant within the temperature range

T ( x, t )          T ( x, t )     2T ( x, t )        
J                                    D                 D
x                    t             x 2              c
Example 1: Rod with thermal reservoir at one
end
δq>0                              δq=0
TR                 T(x,0)=T1<TR

• 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
δq>0                                           δq=0
TR                          T(x,∞)=T1=TR

 u  c T                         U  cV TR  T1 

u                                        TR
s                                 S  cV ln
T                                         T1
Boundary conditions:

T ( x  0, t )  TR   TR                  L ~ Dt
T ( x  L, t )
0    T1
x
Example 2: Rod with thermal reservoirs at
different temperatures at each end

TR1
δq<0                                δq>0 T
TR1<T(x,0)=T1<TR2           R2

• Questions:
– What is the change in energy and entropy of the rod
– 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
TR1<T(x,t)<TR2                R2

• System never reaches equilibrium since there is always a
• Steady-state temperature profile is linear
TR2

TR1

 TR1  TR 2                                           
q   
q
U  cV              T1     
U ss  0   
S ss  0   Stot _ ss     
     2                                                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 ,U 3 ,U 4 ...U s
isolated system
Boltzmann Factor

• Recall:     log  1
          U  T  log 
U     T
• Energy is conserved
U tot  constant            U tot  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

R Utot 
system:

• Number of states of reservoir when in contact with
small system in state γs:
R Utot  Us 
• 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            Thermal
system           Reservoir
1       x x x        x x               1* R U tot  U s 
2
x   Ps 
x       x                  tot
3          x x
x x
x       x
x
4
tot   R Utot  U s 
x
x x               x
x        x
s          x x         x      x          s
Boltzmann Distribution
 Us 
tot   R Utot  U s    &    R U tot  U s   R U tot  exp   
s
 TR 
 Us 
tot   R U tot   exp   
s     TR 
 Us 
• Identify the partition function:            Z   exp   
s     TR 

tot  R Utot  * Z
• Revised expression for probability of state s:
U s
exp 


          TR 

Ps 
Z
Configurations
Small             Thermal
• Frequently interested            system            Reservoir
in a macroscopic               1      x x x     x x      x
property                       2             x       x
3        x x
x       x
• Subset of states of a                   x x x
4         x x
x     x
system called a                                x        x
configuration                  s        x x       x      x

• Probability of a
 Us 
configuration (A) is    Z A   exp   
sum of probability of          As  TR                   ZA
states (s) contained in                               Ps 
 Us                    Z
the configuration       Z   exp   
s       TR 

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