No class Mon 11/7 (ISCB conference)
HW7 due Fri 10/28
quiz MR + Ch 10 Fri 10/28
Boltzmann + F=-kTlnQ
Also no class Fri 11/18 (Thanksgiving)
Exam II Fri 11/11
In a two-particle system, energies of the two particles add, so the partition functions multiply:
Regular solution model:
HW 11 due 12/9
Fmix
= AB x(1 x) + x ln x + (1 x) ln(1 x)
NkT
U mix
can be + or -mole fractions:
x = xA = N A N
1 x = xB = N B N
U mix from lattice model and energy of
AB contacts relative to AA & BB contacts
using mean field approx
Beer Office Hrs: Fri 9/30 10am, Mon 10/3 10am Clark 316
No class Wed 10/5 (MB at NIH)
Exam I Thru Ch 5 Entropy & the Boltzmann Law
Friday Oct 7 in class, 11-11:50am. Calculators & 1pg. notes
Will post practice problems, go over solutions in eve Review ses
HW1 due Fri 9/9 Read Ch 1, 2, and 3
Why probability? We have no hope of fully describing the precise state of a macro-molecular
system. The beauty and power of stat mech is that probabilistic approaches allow us to
describe properties of the system that d
HW9 due 11/25
S = Nk ln
3
2
cU V
N
Ideal Gas
2-state system
5
2
3
Nk ln U
2
cV
+ Nk ln 5 / 2
N
S (U ) =
1 S
3 Nk
=
=
T U V , N 2 U
T=
2
U
3 Nk
f excited
1
k
= ln
T
0 f ground
=
k
0
ln
U
N 0 U
Ideal Gas
T=
U=
2-state system
2
U
3 Nk
3
NkT
2
3
U
CV =
Exam II Fri 11/11 will cover Ch6 - Ch 11:
Exam II Practice Problems posted
Review Session next week
HW9 will be due in 2 weeks
2 2
2
2
2 + 2 + 2 + V ( x, y, z ) = j
i
=
z
t
2m x y
+ V ( x, y, z ) =
+
+
2
2
2
2
z
8 m x y
h
2
2
2
2
( x , t ) = n (
580.321 Statistical Mechanics and Thermodynamics
Fall 2016
Course Description:
This course will introduce the basic principles of statistical physics and thermodynamics as
they apply to biological systems. Topics include:
basic principles of probability
f
Statistical Mechanics:
Ch 6,7 HW5 due 10/14
Thermodynamic systems in maximum entropy state subject to constraints on (U,V,N).
When constraints are released (energy flows, etc) new equilibrium is maximum total entropy
state, subject to constraints on how
HW 3 due Fri
Generalization of Lagrange multiplier technique to functions of t variables: f ( x1 , x2 , , xt )
extrema of
f ( x1 , x2 , , xt ) subject to constraint g ( x1 , x2 , , xt ) = c
t
g
f
dg = dxi = 0
df = dxi = 0
i =1 xi
i =1 xi
t
f
g
dxi = 0
d
Chapter 1
Principles of Probability
1. Combining independent probabilities.
You have applied to three medical schools: University of California at San Francisco (UCSF),
Duluth School of Mines (DSM), and Harvard (H). You guess that the probabilities youll
Quiz 3 Fri (Ch 4)
There is an extensive function S = k ln W
, and we call this quantity the entropy
Because lnW is monotonic, maximizing S maximizes W.
The entropy can also be written in terms of the probability distribution function of a system:
t
entrop
Ch 4 Multivariate calculus:
multivariate functions assign one value to y for each set
of values (x1, x2,xt): y=f(x1, x2,xt)
Partial derivatives:
f ( x + x, y ) f ( x, y )
f
= lim
x
x y x0
f
f ( x, y + y ) f ( x, y )
= lim
y
y x y 0
2 f
2
x
2 f
HW6 posted due Fri 10/21
Quiz Ch8 Thurs 10/20
From fundamental equations:
dU = TdS pdV + dN
p
1
dS = dU + dV dN
T
T
T
dF = SdT pdV + dN
dH = TdS + Vdp + dN
dG = SdT + Vdp + dN
for const Volume and dN=0 process:
dU = TdS = dq
q U
CV (T )
=
dT V T V
or
Exam: Fri 10/7/2016 Maryland 110 11-11:50am
Review Mon 7-9pm, Exam I practice posted
will post practice answers Tues night
Topics: (thru Ch 5)
8.5x11 sheet, CALCULATOR
Basic Probability and Combinatorics
t
Probability Distribution Functions: normalization
Maxwells Relations.
Fundamental Equations:
dU = TdS pdV + dN
dF = SdT pdV + dN
dH = TdS + Vdp + dN
dG = SdT + Vdp + dN
Quiz Thurs, HW6 due Fri
U ( S ,V , N )
F (T , V , N ) = U TS
H ( S , p, N ) = U + pV
G (T , p, N ) = U + pV TS
min at fixed T,V
min at f
Appendix A,C (2nd Ed): Series & Approximations
HW2 due Fri 9/16
n
sn = ax k 1 = a + ax + ax 2 + + ax n 1
geometric series:
k =1
sn x = ax + ax 2 + ax 3 + + ax n
sn sn x = a ax
The infinite series:
s = ax k 1 = a + ax + ax 2 +
converges if |x|<1 to:
k =1
When the total energy, or average energy per particle, is constrained, maximizing S leads to
exponential distributions:
t
t
st
1 constraint: g ( p1 , p2 , pt ) = pi = 1
S = p ln p
i =1
i =1
t
E
constraint: h( p1 , p2 , , pt ) = i pi = =
N
i =1
S
g
h
S
=0
Ch 15. Solutions and mixtures:
HW10 due 12/2, HW11 due 12/9
Quiz Fri 12/2 Ch13,14
The entropy change of mixing NA molecules of type A, and NB molecules of type B is:
S mix = k ln W = k ln
N!
N A! N B !
N = N A + NB
S mix = k ( N ln N N A ln N A N B ln N B
Exam II Fri 11/11 Ch 6-11
HW9 will be due in 2 weeks
Topics:
Fundamental Eqn for Energy U(S,V,N), defs of T,p,
Defs and Fund Eqns for S(U,V,N) ,F(T,V,N),H(S,p,N),G(T,p,N) and others
Lattice Model of Ideal Gas
Maximizing Entropy at fixed U
Minimizing F at
Read Ch 10. The Boltzmann Distribution Law
t
E and
Before, with fixed:
= i pi =
i =1
maximized when
e i
*
pi =
q
t
q = e
N
t
p
i =1
i
HW 7 due 10/28
t
S = k pi ln pi
=1
i =1
determined by :
i
t
t
i =1
i =1
= i pi = i
i =1
e
t
i
q
=
e
i
i =1
t
i
e
Exam: Fri 10/7/2016 Maryland 110 11:00-11:50am
Review Session
HW5 due in ~2 weeks
Topics: (thru Ch 5)
8.5x11 sheet, will post practice problems Thurs night
Basic Probability and Combinatorics
t
Probability Distribution Functions: normalization pi = 1 aver
Read Ch 2. Extremum Principles
HW2 posted, due 9/16 also read Ch. 3
The state of a system is specified by the positions x i and velocities v i of its components.
The dynamics of system specified by Newtons
2nd
In practice, for n > 3 this is difficult, for
Homework #1 Systems Bioengineering I, Fall 2011, page 1 of 7
Homework 1: Ion-channel permeation
(100 points in total)
1. (20 pts) Consider the dimensionally pseudo-realistic permeation pathway shown in the figure below.
It is 40 angstroms long, and the di
Homework XX - Contractile Mechanisms I and II
65 points total = 10 + 31 + 24
1) (10 points total, 1 point each) MUSCLE SECTION, SLIDING FILAMENT THEORY. Use
the following graph for maximally activated steady-state force in skeletal muscle. Striation
spaci
Homework - Contractile Mechanisms Part II: Huxley 57 Derivation and Simulation,
Types of Muscle, and Excel-based Monte Carlo Simulation
Total points:90 = 10 + 35 + 30 + 15
1) (10 points) HUXLEY 57 MODEL DERIVATION. In the class, we defined the probability
Homework 5 Excel-based Huxley 57 Simulation, Types of Muscle, and Excelbased Monte Carlo Simulation
Four problems for 90 points total = 29 + 30 + 20 + 11
1) (Please use the 2008 Excel-based implementation of the Huxley 57 model for this
problem.
a) (4 poi
Homework #2 KEY for Yue section of Systems Bioengineering I, Fall 2011, page 1 of 9
Homework 2 KEY: Ion-channel gating (100 points in total)
1. (40 pts) Consider a simple human voltage-activated K channel with a single n gate, whose
properties are given b
Homework XX - Contractile Mechanisms I and II
65 points total = 10 + 31 + 24
1) (10 points total, 1 point each) MUSCLE SECTION, SLIDING FILAMENT THEORY. Use
the following graph for maximally activated steady-state force in skeletal muscle. Striation
spaci
Homework #1 for Yue section of SBE I, Fall 2009, page 1 of 9
Homework 1: Ion-channel permeation
(100 points in total)
1. (10 pts) Consider the network of BR-permeation channels, placed in series as drawn
below. Let the voltage difference between nodes 1 a