We describe a simple framework for teaching the principles that underlie the dynamical laws of transport: Fick’s law of diffusion, Fourier’s law of heat flow, the Newtonian viscosity law, and the mass-action laws of chemical kinetics. In analogy with the way that the maximization of entropy over microstates leads to the Boltzmann distribution and predictions about equilibria, maximizing a quantity that E. T. Jaynes called “caliber” over all the possible microtrajectories leads to these dynamical laws. The principle of maximum caliber also leads to dynamical distribution functions that characterize the relative probabilities of different microtrajectories. A great source of recent interest in statistical dynamics has resulted from a new generation of single-particle and single-molecule experiments that make it possible to observe dynamics one trajectory at a time.

1.
F.
Reif
,
Fundamentals of Statistical and Thermal Physics
(
McGraw-Hill
,
New York
,
1965
).
2.
K.
Dill
and
S.
Bromberg
,
Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology
(
Garland Science
,
New York
,
2003
).
3.
J.
Kasianowicz
,
E.
Brandin
,
D.
Branton
, and
D.
Deamer
, “
Characterization of individual polynucleotide molecules using a membrane channel
,”
Proc. Natl. Acad. Sci. U.S.A.
93
,
13770
13773
(
1996
).
4.
H. P.
Lu
,
L.
Xun
, and
X. S.
Xie
, “
Single molecule enzymatic dynamics
,”
Science
282
,
1877
1882
(
1998
).
5.
M.
Reif
,
R. S.
Rock
,
A. D.
Mehta
,
M. S.
Mooseker
,
R. E.
Cheney
, and
J. A.
Spudich
, “
Myosin-V stepping kinetics: A molecular model for processivity
,”
Proc. Natl. Acad. Sci. U.S.A.
97
,
9482
9486
(
2000
).
6.
A.
Meller
,
L.
Nivon
, and
D.
Branton
, “
Voltage-driven DNA translocations through a nano-pore
,”
Phys. Rev. Lett.
86
,
3435
3438
(
2001
).
7.
D. E.
Smith
,
S. J.
Tans
,
S. B.
Smith
,
S.
Grimes
,
D. L.
Anderson
, and
C.
Bustamante
, “
The bacteriophage ϕ29 portal motor can package DNA against a large internal force
,”
Nature (London)
413
,
748
752
(
2001
).
8.
H.
Li
,
W. A.
Linke
,
A. F.
Oberhauser
,
M.
Carrion-Vazquez
,
J. G.
Kerkvliet
,
H.
Lu
,
P.
Marszalek
, and
J. M.
Fernandez
, “
Reverse engineering of the giant muscle protein
,”
Nature (London)
418
,
998
1002
(
2002
).
9.
J.
Liphardt
,
S.
Dumont
,
S. B.
Smith
,
I.
Tinocho
, and
C.
Bustamante
, “
Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality
,”
Science
296
,
1832
1835
(
2002
).
10.
C.
Bustamante
,
Z.
Bryant
, and
S. B.
Smith
, “
Ten Years of tension: single-molecule DNA mechanics
,”
Nature (London)
421
,
423
427
(
2003
).
11.
E. R.
Dufresne
,
D.
Altman
, and
D. G.
Grier
, “
Brownian dynamics of sphere in a slit pore
,”
Europhys. Lett.
53
,
264
270
(
2001
).
12.
B.
Alberts
,
D.
Bray
,
A.
Johnson
,
J.
Lewis
,
M.
Raff
,
K.
Roberts
, and
P.
Walter
,
Essential Cell Biology: An Introduction to the Molecular Biology of the Cell
(
Garland
,
New York
,
1997
).
13.
D.
Kondepudi
and
I.
Prigogine
,
Modern Thermodynamics: From Heat Engines to Dissipative Structures
(
Wiley
,
New York
,
1998
).
14.
E. T.
Jaynes
, in
E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics
, edited by
R. D.
Rosenkrantz
(
Kluwer Academic
,
1980
), Chap. 14.
15.
M.
Klein
, “
Entropy and the Ehrenfest urn model
,”
Physica (Amsterdam)
22
,
569
575
(
1956
).
16.
V.
Ambegaokar
and
A.
Clerk
, “
Entropy and time
,”
Am. J. Phys.
67
,
1068
1073
(
1999
).
17.
G. G.
Emch
and
C.
Liu
,
The Logic of Thermostatistical Physics
(
Springer-Verlag
,
New York
,
2002
).
18.
R. P.
Feynman
in
Nobel Lectures in Physics: 1901–1995
(
World Scientific
,
Singapore
,
1998
).
19.
Reference 14, Chap. 1.
20.
M.
Kac
, “
Random walk and the theory of Brownian motion
,”
Am. Math. Monthly
54
,
369
391
(
1947
).
21.

The Maxwell demon is an imaginary microscopic object that was invoked in similar situations in heat flow processes, that is, where heat flows from a colder object to a hotter one, albeit with low probability (Ref. 22). In particular, the demon is supposed to capture the bad actor microtrajectories.

22.
G.
Gamow
,
One Two Three … Infinity
(
Dover
,
New York
,
1988
).
23.
G. M.
Wang
,
E. M.
Sevick
,
E.
Mittag
,
D. J.
Searles
, and
D. J.
Evans
, “
Experimental verification of the fluctuation theorem
,”
Phys. Rev. Lett.
89
,
050601
1
(
2002
).
24.
J.
Howard
,
Mechanics of Motor Proteins and the Cytoskeleton
(
Sinauer Associates
,
2001
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.