Anomalous diffusion in generalised Ornstein-Uhlenbeck processes

1. Anomalous diffusion in generalised Ornstein-Uhlenbeck processes We investigated anomalous (and regular) diffusion in a physical process which does not contain power laws in its microscopic description. The problem is analysed by constructing the eigenfunctions and eigenvalues of a Fokker-Planck operator. The spectrum consists of two staggered ladders, and the states are generated using annhilation and creation operators which contain second derivatives. M. Wilkinson, B. Mehlig, V. Bezuglyy, K. Nakamura, E. Arvedson Staggered Ladder Spectra,E. Arvedson, M. Wilkinson, B. Mehlig and K. Nakamura,Phys. Rev. Lett., 96, 030601, (2006). Generalised Ornstein-Uhlenbeck Processes,V. Bezuglyy, B. Mehlig, M. Wilkinson, K. Nakamura and E. Arvedson,J. Math. Phys., in press, (2006).

2. The Ornstein-Uhlenbeck process Short correlation time: use Langevin approximation Second moments show diffusive behaviour:

3. Generalised Ornstein-Uhlenbeck processes Consider a force which depends upon x as well as t: Correlation length is x, correlation time t. If , the force experienced by the particle decorrelates much more rapidly than for a stationary particle. Impulse supplied by random force in time Dt is: For estimate The diffusion constant D(p) decreases as p increases:

4. Fokker-Planck equation The probability density for momentum, P(p,t), satisfies a Fokker-Planck equation: We found exact solutions when The steady-state solution is non-Maxwellian: Change to dimensionless variables , : It is useful to transform the Fokker-Planck operator to a Hermitean form (the ‘Hamiltonian’):

5. Ladder operators and eigenfunctions Consider the eigenfunctions and eigenvalues, satisfying: The first two of each parity are found by inspection: It is convenient to introduce various operators: The most significant identities are We deduce that the spectrum consists of two staggered ladders:

6. Staggered ladder spectrum

7. Propagator and expectation values The spectral decomposition is used to construct the probability propagator from y to z in time t’ (dimensionless variables): Expectation value of O(z) (starting from z=0 at t’=0): Correlation function of O(z) (starting from z=0 at t’=0):

8. Some results By using the commutation relations of the raising and lowering operators, it is possible to find recursion relations for matrix elements. Expectation values require ratios of wavefunctions, also fond by a recursion relation. Using these relations, we can determine the second moment of the momentum:

9. Spatial diffusion Express momentum correlation function in terms of spectrum: Sum rule and asymptotic form for matrix elements: At short times, many terms contribute: approximate sums by integrals, find

10. Growth of second moments

11. Gradient-force case and generalisation When , find In general case, we consider Hamiltonian operator and ground state: Operator relations and spectrum: Anomalous diffusion for short times in general case:

12. Conclusions There are very few systems with ladder spectra (harmonic oscillator, Zeeman splitting). We have found a new example. The ladder spectrum is analysed by means of raising and lowering operators which are second order differential operators. These operators enable matrix elements to be evaluated by recursion. The generalised Ornstein-Uhlenbeck system exhibits anomalous diffusion for times such that gt is small. Generic random force: Random potential: There are no fractional power-laws in the microscopic definition of the model. Earlier work: Momentum-dependent diffusion constant: Anomalous exponents for the potential case, for model with g=0: