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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
High Temperature
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
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