# Kramers’ Theory

• Philipp Scherer
• Sighart F. Fischer
Chapter
Part of the Biological and Medical Physics, Biomedical Engineering book series (BIOMEDICAL)

## Abstract

Kramers [32] used the concept of Brownian motion to describe motion of particles over a barrier as a model for chemical reactions in solution. The probability distribution of a particle moving in an external potential is described by the Klein–Kramers equation (7.71):
$$\begin{array}{c} \frac{{\partial W(x,v,t)}}{{\partial t}} = \left[ - {\frac{\partial } {{\partial x}}v + \frac{\partial } {{\partial v}}\left( {\gamma v - \frac{{K(x)}} {m}}\right)\, + \,\frac{{\gamma K_B T}} {m}\frac{{\partial ^2 }}{{\partial v^2 }}} \right]W(x,v,t) \\ = \frac{\partial } {{\partial x}}S_x + \frac{\partial } {{\partial v}}Sv \\ \end{array}$$
(8.1)
and the rate of the chemical reaction is related to the probability current Sx across the barrier.

## Keywords

Partition Function Brownian Motion Stationary Solution Saddle Point Harmonic Oscillator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
T.L. Hill, An Introduction to Statistical Thermodynamics (Dover, New York, 1986)Google Scholar
2. 2.
S. Cocco, J.F. Marko, R. Monasson, arXiv:cond-mat/0206238v1Google Scholar
3. 3.
C. Storm, P.C. Nelson, Phys. Rev. E 67, 51906 (2003)
4. 4.
K.K. Mueller-Niedebock, H.L. Frisch, Polymer 44, 3151 (2003)Google Scholar
5. 5.
C. Leubner, Eur. J. Phys. 6, 299 (1985)
6. 6.
P.J. Flory, J. Chem. Phys. 10, 51 (1942)
7. 7.
M.L. Huggins, J. Phys. Chem. 46, 151 (1942)Google Scholar
8. 8.
M. Feig, C.L. Brooks III, Curr. Opin. Struct. Biol. 14, 217 (2004)Google Scholar
9. 9.
B. Roux, T. Simonson, Biophys. Chem. 78, 1 (1999)Google Scholar
10. 10.
A. Onufriev, Annu. Rep. Comput. Chem. 4, 125 (2008)Google Scholar
11. 11.
M. Born, Z. Phys. 1, 45 (1920)
12. 12.
D. Bashford, D. Case, Annu. Rev. Phys. Chem. 51, 29 (2000)Google Scholar
13. 13.
W.C. Still, A. Tempczyk, R.C. Hawley, T. Hendrickson, JACS 112, 6127 (1990)Google Scholar
14. 14.
G.D. Hawkins, C.J. Cramer, D.G. Truhlar, Chem. Phys. Lett. 246, 122 (1995)
15. 15.
M. Schaefer, M. Karplus, J. Phys. Chem. 100, 1578 (1996)Google Scholar
16. 16.
I.M. Wonpil, M.S. Lee, C.L. Brooks III, J. Comput. Chem. 24, 1691 (2003)Google Scholar
17. 17.
F. Fogolari, A. Brigo, H. Molinari, J. Mol. Recognit. 15, 377 (2002)Google Scholar
18. 18.
A.I. Shestakov, J.L. Milovich, A. Noy, J. Colloid Interface Sci. 247, 62 (2002)Google Scholar
19. 19.
B. Lu, D. Zhang, J.A. McCammon, J. Chem. Phys. 122, 214102 (2005)
20. 20.
P. Koehl, Curr. Opin. Struct. Biol. 16, 142 (2006)Google Scholar
21. 21.
P. Debye, E. Hueckel, Phys. Z. 24, 305 (1923)Google Scholar
22. 22.
G. Goüy, Comt. Rend. 149, 654 (1909)Google Scholar
23. 23.
G. Goüy, J. Phys. 9, 457 (1910)Google Scholar
24. 24.
D.L. Chapman, Philos. Mag. 25, 475 (1913)Google Scholar
25. 25.
O. Stern, Z. Elektrochem. 30, 508 (1924)Google Scholar
26. 26.
A.-S. Yang, M. Gunner, R. Sampogna, K. Sharp, B. Honig, Proteins 15, 252 (1993)Google Scholar
27. 27.
P.W. Atkins, Physical Chemistry (Freeman, New York, 2006)Google Scholar
28. 28.
W.J. Moore, Basic Physical Chemistry (Prentice-Hall, New York, 1983)Google Scholar
29. 29.
M. Schaefer, M. Sommer, M. Karplus, J. Phys. Chem. B 101, 1663 (1997)Google Scholar
30. 30.
R.A. Raupp-Kossmann, C. Scharnagl, Chem. Phys. Lett. 336, 177 (2001)
31. 31.
H. Risken, The Fokker–Planck Equation (Springer, Berlin, 1989)
32. 32.
H.A. Kramers, Physica 7, 284 (1941)
33. 33.
P.O.J. Scherer, Chem. Phys. Lett. 214, 149 (1993)
34. 34.
E.W. Montroll, H. Scher, J. Stat. Phys. 9, 101 (1973)
35. 35.
E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)
36. 36.
A.A. Zharikov, P.O.J. Scherer, S.F. Fischer, J. Phys. Chem. 98, 3424 (1994)Google Scholar
37. 37.
A.A. Zharikov, S.F. Fischer, Chem. Phys. Lett. 249, 459 (1995)
38. 38.
S.R. de Groot, P. Mazur, Irreversible Thermodynamics (Dover, New York, 1984)Google Scholar
39. 39.
40. 40.
41. 41.
D.E. Goldman, J. Gen. Physiol. 27, 37 (1943)Google Scholar
42. 42.
A.L. Hodgkin, A.F. Huxley, J. Physiol. 117, 500 (1952)Google Scholar
43. 43.
J. Kenyon, How to solve and program the Hodgkin–Huxley equations (http://134.197.54.225/department/Faculty/kenyon/Hodgkin&Huxley/pdfs/HH.Program.pdf)
44. 44.
J. Vreeken, A friendly introduction to reaction–diffusion systems, Internship Paper, AILab, Zurich, July 2002Google Scholar
45. 45.
E. Pollak, P. Talkner, Chaos 15, 026116 (2005)
46. 46.
F.T. Gucker, R.L. Seifert, Physical Chemistry (W.W. Norton, New York, 1966)Google Scholar
47. 47.
S. Glasstone, K.J. Laidler, H. Eyring, The Theory of Rate Processes (McGraw-Hill, New York, 1941)Google Scholar
48. 48.
K.J. Laidler, Chemical Kinetics, 3rd edn. (Harper & Row, New York, 1987)Google Scholar
49. 49.
G.A. Natanson, J. Chem. Phys. 94, 7875 (1991)
50. 50.
R.A. Marcus, Annu. Rev. Phys. Chem. 15, 155 (1964)
51. 51.
R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811, 265 (1985)Google Scholar
52. 52.
R.A. Marcus, Angew. Chem. Int. Ed. Engl. 32, 1111 (1993)Google Scholar
53. 53.
A.M. Kuznetsov, J. Ulstrup, Electron Transfer in Chemistry and Biology (Wiley, Chichester, 1998), p. 49Google Scholar
54. 54.
P.W. Anderson, J. Phys. Soc. Jpn. 9, 316 (1954)
55. 55.
R. Kubo, J. Phys. Soc. Jpn. 6, 935 (1954)
56. 56.
R. Kubo, in Fluctuations, Relaxations and Resonance in Magnetic Systems, ed. by D. ter Haar (Plenum, New York, 1962)Google Scholar
57. 57.
T.G. Heil, A. Dalgarno, J. Phys. B 12, 557 (1979)
58. 58.
L.D. Landau, Phys. Z. Sowjetun. 1, 88 (1932)
59. 59.
C. Zener, Proc. R. Soc. Lond. A 137, 696 (1932)
60. 60.
E.S. Kryachko, A.J.C. Varandas, Int. J. Quant. Chem. 89, 255 (2001)Google Scholar
61. 61.
E.N. Economou, Green’s Functions in Quantum Physics (Springer, Berlin, 1978)Google Scholar
62. 62.
H. Scheer, W.A. Svec, B.T. Cope, M.H. Studler, R.G. Scott, J.J. Katz, JACS 29, 3714 (1974)Google Scholar
63. 63.
A. Streitwieser, Molecular Orbital Theory for Organic Chemists (Wiley, New York, 1961)Google Scholar
64. 64.
J.E. Lennard-Jones, Proc. R. Soc. Lond. A 158, 280 (1937)
65. 65.
B.S. Hudson, B.E. Kohler, K. Schulten, in Excited States, ed. by E.C. Lin (Academic, New York, 1982), pp. 1–95Google Scholar
66. 66.
B.E. Kohler, C. Spangler, C. Westerfield, J. Chem. Phys. 89, 5422 (1988)
67. 67.
T. Polivka, J.L. Herek, D. Zigmantas, H.-E. Akerlund, V. Sundstrom, Proc. Natl. Acad. Sci. USA 96, 4914 (1999)
68. 68.
B. Hudson, B. Kohler, Synth. Met. 9, 241 (1984)Google Scholar
69. 69.
B.E. Kohler, J. Chem. Phys. 93, 5838 (1990)
70. 70.
W.T. Simpson, J. Chem. Phys. 17, 1218 (1949)
71. 71.
H. Kuhn, J. Chem. Phys. 17, 1198 (1949)
72. 72.
M. Gouterman, J. Mol. Spectrosc. 6, 138 (1961)
73. 73.
M. Gouterman, J. Chem. Phys. 30, 1139 (1959)
74. 74.
M. Gouterman, G.H. Wagniere, L.C. Snyder, J. Mol. Spectrosc. 11, 108 (1963)
75. 75.
C. Weiss, The Porphyrins, vol. III (Academic, New York, 1978), p. 211Google Scholar
76. 76.
D. Spangler, G.M. Maggiora, L.L. Shipman, R.E. Christofferson, J. Am. Chem. Soc. 99, 7470 (1977)Google Scholar
77. 77.
B.R. Green, D.G. Durnford, Annu. Rev. Plant Physiol. Plant Mol. Biol. 47, 685 (1996)Google Scholar
78. 78.
H.A. Frank et al., Pure Appl. Chem. 69, 2117 (1997)Google Scholar
79. 79.
R.J. Cogdell et al., Pure Appl. Chem. 66, 1041 (1994)Google Scholar
80. 80.
H. van Amerongen, R. van Grondelle, J. Phys. Chem. B 105, 604 (2001)Google Scholar
81. 81.
T. Frster, Ann. Phys. 2, 55 (1948)Google Scholar
82. 82.
83. 83.
D.L. Dexter, J. Chem. Phys. 21, 836 (1953)
84. 84.
F.J. Kleima, M. Wendling, E. Hofmann, E.J.G. Peterman, R. van Grondelle, H. van Amerongen, Biochemistry 39, 5184 (2000)Google Scholar
85. 85.
T. Pullerits, M. Chachisvilis, V. Sundstrm, J. Phys. Chem. 100, 10787 (1996)Google Scholar
86. 86.
R.C. Hilborn, Am. J. Phys. 50, 982 (1982), revised 2002Google Scholar
87. 87.
S.H. Lin, Proc. R. Soc. Lond. A 335, 51 (1973)
88. 88.
S.H. Lin, W.Z. Xiao, W. Dietz, Phys. Rev. E 47, 3698 (1993)
89. 89.
MOLEKEL 4.0, P. Fluekiger, H.P. Luethi, S. Portmann, J. Weber, Swiss National Supercomputing Centre CSCS, Manno Switzerland, 2000Google Scholar
90. 90.
J. Deisenhofer, H. Michel, Science 245, 1463 (1989)
91. 91.
J. Deisenhofer, O. Epp, K. Miki, R. Huber, H. Michel, Nature 318, 618 (1985)
92. 92.
J. Deisenhofer, O. Epp, K. Miki, R. Huber, H. Michel, J. Mol. Biol. 180, 385 (1984)Google Scholar
93. 93.
H. Michel, J. Mol. Biol. 158, 567 (1982)Google Scholar
94. 94.
E.W. Knapp, P.O.J. Scherer, S.F. Fischer, BBA 852, 295 (1986)Google Scholar
95. 95.
P.O.J. Scherer, S.F. Fischer, in Chlorophylls, ed. by H. Scheer (CRC, Boca Raton, 1991), pp. 1079–1093Google Scholar
96. 96.
R.J. Cogdell, A. Gall, J. Koehler, Q. Rev. Biophys. 39, 227 (2006)Google Scholar
97. 97.
M. Ketelaars et al., Biophys. J. 80, 1591 (2001)
98. 98.
M. Matsushita et al., Biophys. J. 80, 1604 (2001)
99. 99.
K. Sauer, R.J. Cogdell, S.M. Prince, A. Freer, N.W. Isaacs, H. Scheer, Photochem. Photobiol. 64, 564 (1996)Google Scholar
100. 100.
A. Freer, S. Prince, K. Sauer, M. Papitz, A. Hawthorntwaite-Lawless, G. McDermott, R. Cogdell, N.W. Isaacs, Structure 4, 449 (1996)Google Scholar
101. 101.
M.Z. Papiz, S.M. Prince, A. Hawthorntwaite-Lawless, G. McDermott, A. Freer, N.W. Isaacs, R.J. Cogdell, Trends Plant Sci. 1, 198 (1996)Google Scholar
102. 102.
G. McDermott, S.M. Prince, A. Freer, A. Hawthorntwaite-Lawless, M. Papitz, R. Cogdell, Nature 374, 517 (1995)
103. 103.
N.W. Isaacs, R.J. Cogdell, A. Freer, S.M. Prince, Curr. Opin. Struct. Biol. 5, 794 (1995)Google Scholar
104. 104.
E.E. Abola, F.C. Bernstein, S.H. Bryant, T.F. Koetzle, J. Weng, in Crystallographic Databases – Information Content, Software Systems, Scientific Applications, ed. by F.H. Allen, G. Bergerhoff, R. Sievers (Data Commission of the International Union of Crystallography, Cambridge, 1987), p. 107Google Scholar
105. 105.
F.C. Bernstein, T.F. Koetzle, G.J.B. Williams, E.F. Meyer Jr., M.D. Brice, J.R. Rodgers, O. Kennard, T. Shimanouchi, M. Tasumi, J. Mol. Biol. 112, 535 (1977)Google Scholar
106. 106.
R. Sayle, E.J. Milner-White, Trends Biochem. Sci. 20, 374 (1995)Google Scholar
107. 107.
Y. Zhao, M.-F. Ng, G.H. Chen, Phys. Rev. E 69, 032902 (2004)
108. 108.
A.M. van Oijen, M. Ketelaars, J. Khler, T.J. Aartsma, J. Schmidt, Science 285, 400 (1999)Google Scholar
109. 109.
C. Hofmann, T.J. Aartsma, J. Khler, Chem. Phys. Lett. 395, 373 (2004)
110. 110.
S.E. Dempster, S. Jang, R.J. Silbey, J. Chem. Phys. 114, 10015 (2001)
111. 111.
S. Jang, R.J. Silbey, J. Chem. Phys. 118, 9324 (2003)
112. 112.
K. Mukai, S. Abe, Chem. Phys. Lett. 336, 445 (2001)
113. 113.
R.G. Alden, E. Johnson, V. Nagarajan, W.W. Parson, C.J. Law, R.G. Cogdell, J. Phys. Chem. B 101, 4667 (1997)Google Scholar
114. 114.
V. Novoderezhkin, R. Monshouwer, R. van Grondelle, Biophys. J. 77, 666 (1999)Google Scholar
115. 115.
M.K. Sener, K. Schulten, Phys. Rev. E 65, 31916 (2002)
116. 116.
A. Warshel, S. Creighton, W.W. Parson, J. Phys. Chem. 92, 2696 (1988)Google Scholar
117. 117.
M. Plato, C.J. Winscom, in The Photosynthetic Bacterial Reaction Center, ed. by J. Breton, A. Vermeglio (Plenum, New York, 1988), p. 421Google Scholar
118. 118.
P.O.J. Scherer, S.F. Fischer, Chem. Phys. 131, 115 (1989)
119. 119.
L.Y. Zhang, R.A. Friesner, Proc. Natl. Acad. Sci. USA 95, 13603 (1998)
120. 120.
M. Gutman, Structure 12, 1123 (2004)Google Scholar
121. 121.
P. Mitchell, Biol. Rev. Camb. Philos. Soc. 41, 445 (1966)Google Scholar
122. 122.
H. Luecke, H.-T. Richter, J.K. Lanyi, Science 280, 1934 (1998)
123. 123.
R. Neutze et al., BBA 1565, 144 (2002)Google Scholar
124. 124.
D. Borgis, J.T. Hynes, J. Chem. Phys. 94, 3619 (1991)
125. 125.
F. Juelicher, in Transport and Structure: Their Competitive Roles in Biophysics and Chemistry, ed. by S.C. Mller, J. Parisi, W. Zimmermann. Lecture Notes in Physics (Springer, Berlin, 1999)Google Scholar
126. 126.
A. Parmeggiani, F. Juelicher, A. Ajdari, J. Prost, Phys. Rev. E 60, 2127 (1999)
127. 127.
F. Jlicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997)
128. 128.
F. Jlicher, J. Prost, Progr. Theor. Phys. Suppl. 130, 9 (1998)
129. 129.
H. Qian, J. Math. Chem. 27, 219 (2000)
130. 130.
M.E. Fisher, A.B. Kolomeisky, arXiv:cond-mat/9903308v1Google Scholar
131. 131.
N.D. Mermin, J. Math. Phys. 7, 1038 (1966)
132. 132.
M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.J. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S. Su, T.L. Windus, M. Dupuis, J.A. Montgomery J. Comput. Chem. 14, 1347 (1993)Google Scholar