Langevin’s Equation of Motion: Dynamical States and Stochastic Resonance

In this present work, we numerically observe the existence of two dynamical states of trajectories in the underdamped motion of a particle in an inhomogeneous periodic bistable potential with high friction coefficient driven by a periodic force. The system is inhomogeneous in the sense that the coefficient of friction is periodic in space and with phase difference 'ϕ' with respect to the periodic bistable potential. Just like in sinusoidal potentials, the occurrence of stochastic resonance (SR) in the periodic bistable potential is due to the transitions between the dynamical states at an optimal (thermal) noise strength.


Introduction
Stochastic resonance (SR) from the word stochastic itself refers to a randomness which cannot be determined precisely and resonance means resonate when the frequencies is closed to the natural frequency of the potential barrier.SR originally developed as an explanation for periodic ice age recurrence by Roberto Benzi in 1981[1], and mostly occur in bistable system V(x) = − ax 2 /2 + bx 4 /4, which has two stable states, these two stable states is separated by a barrier at x=0.In this case the system which can be a mutual effect between internal system and external force such effect can be called stochastic resonance.In case of sinusoidal potential we study the two dynamical stable states of a particle trajectories x(t) which differ from each other with a phase lag ϕ = ϕ1 and ϕ = ϕ2, the inter-well transition between the two states at a finite noise strength (T) is stochastic in nature [2].A system like two stable states like ice ages, cold and warm interglacial period.A weak periodic force assisted by random fluctuation noise can make the particle transition between the two dynamical states in the system.But this ideas was not appropriate for explanation of ice ages.The mutual effect of fluctuation (noise) and external periodic force got a suitable name called stochastic resonance.Stochastic resonance (SR) can be studied in periodic potentials.The possibility of SR to occur in a periodic manner is that a system which is driven by a periodic force has to be a tilted periodically in that the motions are allowed in one direction only [3].
In this work we discuss the dynamical states of trajectories and SR in periodic bistable potential U(x) = 2/3(cos(x) + cos (2x)) as in Fig. 1 with two sub-well in a well of a potential driven by a periodic force F(t) = F0 cos(ωt) and the asymmetry of the system is brought in with a non-uniform friction γ(x) = γ0(1-λcos(ωt +ϕ), with high friction coefficient γ0 and the inhomogeneity λ, 0 ≤ λ ≤ 1.The periodic bistable potential is important in the study of superionic conductor [4].We used the total input energy and hysteresis loop area (HLA) as a quantifier of SR.HLA gives effective information regarding phase relationship of the position (x) and applied force (F).The input energy is applied to the system shows a peak as a function of noise strength in a same way as HLA, this indicate the occurrence of SR in periodic bistable potential [2,3].
For some interestingly evidence of SR in periodic bistable potential we choose the value of ϕ = 0.0π and 1.0π with minimum asymmetry.The same for the numerical results for trajectories and the absorbed dynamical states for different noise strength and the value of 'τ'.This work we have classified in different section, Sec 2. The theoretical model of our work, Sec 3. Numerical Results based on the above model and Sec 4. Conclusion and summarized.

Numerical Results
Solving equation (2), the particle trajectories x(t) corresponding to a periodic bistable potential with the amplitude of the applied force F0 = 0.15, ω = 2π/τ and the period τ = 4.7, where ω is closed to the natural frequency of the system.The trajectories x(t) for the initial condition (x(0),v(0)) can be obtained numerically [5] and the total input energy w and HLA of the system is numerically solved [6,8].From Figure 2. The trajectories of the particle at a finite temperatures (T=0.001) as a response to the applied force F(t), and the friction coefficient γ (x) = γ0[1 − λ cos(x + ϕ)] is measured by λ and the phase shift with U(x).These trajectories with two stable states of in-phase and out-of-phase trajectories or simple (small and large amplitude) depending on the initial position x(0) and is periodic with the applied force F(t).The input energy  ̅ band average over the trajectory of every initial position x(0) at the lowest temperature T=0.001 Fig 3(a).The two narrow band of  ̅ arround 0.025 and 0.3 and no point lie between the two energy band. ̅ = 0.025 Correspond to the in-phase and  ̅ = 0.30 correspond to the out of phase statesand the energy gap is ~0.275.When the temperature increase the transition start to take place between the inphase and out of phase Figure 3  The transition between the two dynamical states start when the noise strength, temperature T=0.003 for ϕ =0 π and transition start to take place temperature T=0.002 for ϕ =1.0π as in Figure 3. Figure 4 show the transition between the in phase and the out of phase, at certain noise strength T~0.1 the input energy reduce gradually and the transition from out-of-phase to the in-phase states is more frequent.
The magnitude of hysteresis loop area <  ̅ > is equal to magnitude of <  ̅ >.Therefore, the hysteresis loop itself gives the information of total input energy.The amplitude F0 of the applied force F(t) is small, one may expect the position x(t) shows a linear variation with F(t): x(t) = x0 cos(ωt + ϕ).Hence, the hysteresis loop < ̅ (()) > is closely resemble an ellipse as shown in Figure 5 and the average phase difference ϕ is measured from the resulting trajectories.The average total input energy <  ̅ > as a function of thermal noise T is shown in Figures 5 and 6. <  ̅ > shows the peak at T ~ 0.05 and T ~ 0.01 for 4.6 < τ ≤ 4.9.This shows the SR for periodic bistable potential [7]. Figure 6 shows the <  ̅ > as a function of T for τ = 4.625, 4.669, 4.7305 and 4.9, F0 = 0.15, γ0 = 0.10, λ = 0.9, and θ = 1.0π.

Conclusion
A particle in a periodic bistable potential driven by a small applied periodic force in presence of time dependent friction with phase difference 'ϕ' shows the two dynamical states and stochastic resonance (SR).In Figure 3 the average energy  ̅ out-of-phase is larger than the in-phase states and the transitions is took place between the two states.The total average input energy <  ̅ > and the Hysteresis loop area <  ̅ > show the qualified for SR when the thermal noise T is equal to the stochastic resonance temperature TSR. Figure 6 gives us the conclusion of SR curve for the range of τ, 40% ≤ τ ≤ 50% as the best resonance curve.

Acknowledgments
We thank the Computer Centre, North-Eastern Hill University, Shillong, for providing the highperformance computing facility, SULEKOR.Department of Physics, NIT-Meghalaya for the

Figure 2 .
Figure2.Plot of Particle trajectories x(t) and applied force F(t) as a function of time t at T= 0.001, the green line is for the in-phase with phase lag ϕ1 with x(0)= 0.50π, the violet line is the out of phase with phase lag ϕ2 and x(0)= 0.0π with respect to the F(t) blue line.This show the two dynamical state of trajectories.
Figure 3(d)(e)(f) show the transition between the dynamical states for ϕ=1.0π.