Diffusion of Interacting Particles in One dimension

1. Diffusion of Interacting Particles in One dimension Deepak Kumar School of Physical Sciences Jawaharlal Nehru University New Delhi IITM, Chennai Nov. 9, 2008

2. Outline • Introduction and History • Single Particle Diffusion: Role of Boundary conditions • Two-Particle Problem • Bethe’s Ansatz: N-Particle Solution • Tagged Particle Diffusion • Correlations • Applications • Reference: Phys. Rev. E 78, 021133 (2008)

3. Introduction • The concept of ‘Single File Diffusion’ was introduced in a biological context to describe flow of ions through channels in a cell membrane. • These channels are crowded and narrow so that the ions diffuse effectively in one dimension and cannot go past each other. • The lattice version of the problem was first considered by T. E. Harris (J. Appl. Probability 2, 323 (1965)

4. History • Harris showed that the hard-core interaction introduces a qualitative new feature in the diffusion of particles in one dimension. • Mean square displacement • This result received a lot of attention, and has been derived using a number of physical arguments. • Notably, Levitt used the exact methods of one-dimensional classical gas to obtain this result. Phys. Rev. A 8, 3050 (1973)

5. Earlier Work • Numerical studies of the problem also showed the sub-diffusive behavior of type under the condition of constant density of particles. • (P. M. Richard, Phys. Rev. B 16, 1393, 1977; H. van Beijeren et al., Phys. Rev. B 28, 5711, 1983) Now there are some exact results. Rödenbeck et al., (Phys. Rev. E 57, 4382, 1998) obtained the one-particle distribution function for a nonzero density by averaging over initial positions. They obtained the above behavior.

6. Earlier Work • Ch. Aslangul(Europhys. Lett. 44, 284, 1998) gave the exact solution for N particles on a line with one initial condition: all particles are at one point at t=0. • Here we give an exact solution for arbitrary initial conditions. We calculate one particle moments and two-particle correlation functions as expansion in powers of .

7. Single Particle Solution

8. Single Particle Diffusion

9. Two Particle Solution

10. Two Particle Solution

11. Two Vicious Walkers

12. N-Particle Solution

13. N-Particle Distribution Function

14. Tagged Particle Diffusion

15. Large Time Expansion

16. Mean Displacement

17. Mean Displacement

18. Mean Square Displacement

19. Correlations

20. Correlations

21. Correlations: Central Particle to Others

22. Correlations: End Particle to Others

23. An Open Problem The N particle solution obtained by us and Aslangul shows that the one-particle moments behave as but the coefficients vanish as N tends to infinity. It is not clear what emerges in the infinite N limit. Properly one should take a finite line and the go over to nonzero density limit. However, in the present calculations some further conditions like constant density or averaging over initial conditions are imposed, to obtain the sub-diffusive behavior.

24. Experiments • Diffusion of colloidal particles has been studied in one-dimensional channels constructed by photolithography (Wei et al., Science 287, 625,2000; Lin et al., Phys. Rev. Lett. 94, 216001, 2005)and by optical tweezers (Lutz et al., Phys. Rev. Lett. 93, 026001, 2004). • Diffusion of water molecules through carbon nanotubes(Mukharjee et al., Nanosci. Nanotechnol. 7, 1, 2007) • The experiments track the trajectories of single particles and show a transition from normal behaviour at short times to sub-diffusive behavior at large times.

25. Applications: Single File Diffusion Biological Applications • Flow of ions and water through molecular-sized channels in membranes. 2. Sliding proteins along DNA 3. Collective behaviour of biological motors Physical and Chemical Applications 4. Transport of adsorbate molecules through pores in zeolites 5. Carrier migration in polymers and superionic conductors 6. Particle flows in microfluidic devices 7. Migration of adsorbed molecules on surfaces 8. Highway traffic flows

26. Thank You

27. History This problem was first investigated on a linear lattice by T. E. Harris. (J. Appl. Prob. 2, 323, 1965). He obtained a qualitatively nontrivial and important result, i.e. subdiffusivebehaviour of a tagged particle. He derived the result for an infinite number of particles on an infinite lattice with finite density. Many workers rederived this result in many ways and checked it numerically for systems with uniform density. Some experiments have investigated the diffusion of colloidal particle through 1D channels created by photolithography or optical tweezers. Another experiment has studied the water diffusion through carbon nanotubes. There is good support for the subdiffusivebehaviour.

28. Random Thoughts Life as a random walk Embrace Randomness Thank you

29. Diffusion of Interacting Particles in One Dimension Outline 1. Random Walk and Diffusion 2. Boundary Conditions: Method of Images 3. Two Interacting Particles on a Line 4. N Interacting Particles: Bethe’s Ansatz 5. Tagged Particle Diffusion 6. Correlations in Non-equilibrium Assembly 7. Physical Applications Reference: Phys. Rev. E 78, 021133 (2008)

30. Random Walk and Diffusion A particle jumps in each step a distance ‘a’ to the right or to the left on a line with equal probability. Displacement X after N steps

31. Single Particle Solution

32. Two Particle Solution

33. Two Particle Solution

34. Two Particle Solution

35. Two Particle Solution