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Stochastic Resonance in 3D Ising Ferromagnets

Finite 3D Ising ferromagnets are studied in periodic magnetic fields both by computer simulations and mean-field theoretical approaches. The phenomenon of stochastic resonance is revealed. The characteristic peak obtained for the correlation function between the external oscillating magnetic field and magnetization versus the temperature of the system, is studied for various external fields and lattice sizes. Excellent agreement between simulation and theoretical results are obtained.


Introduction
It is well-known [1][2][3][4], that periodically modulated bistable systems in the presence of noise exhibits the phenomenon of stochastic resonance (SR). The bases of this phenomenon is that the correlation σ between the modulation signal and the response of the system presents an extremum for a given noise intensity.
Considering a special and practically important case of coupled two-state systems, recently we reported on the possibility of obtaining the SR in finite two dimensional Ising systems [13]. In the mentioned work, in contrast with all other earlier works we did not consider any external stochastic forces, only the thermal fluctuations in the system. The problem was studied by computer simulations considering a heat-bath dynamics.
The present paper intends to complete the earlier one [13], studying the important three dimensional (3D) case. Presuming that in 3D the mean-field theories give reasonable results, we also consider a theoretical approach to the phenomenon. We discuss and compare our theoretical and computer simulation results.

The Method
For detecting SR we need a system in a double well potential, governed by a stochastic force. In the meantime the two minima of the double well potential must be modulated periodically in antiphase. One can immediately recognize that ferromagnetic Ising systems in oscillating magnetic fields satisfy all these conditions: • at zero thermodynamic temperature the free-energy versus magnetization curve has the double well form, • an external periodic magnetic field [B(t)], modulates the two minima in antiphase, • the effect of a positive temperature can be viewed as a stochastic driving force, • the magnetization as a function of time (M(t) = i S z i ) can be considered as the response function of the system.
Due to the fact that in the proposed system the noise intensity (thermal fluctuations) is temperature dependent, the characteristic maximum for SR must be detected at a resonance temperature (T r ) in the plot of the correlation σ (σ =|< B(t) · M(t) >|) versus the temperature (T).

The Hamiltonian of our problem is
where the sum is referring to all nearest neighbours, S z i = ±1, µ B is the Bohr magneton and B(t) the external magnetic field. We consider B(t) in a harmonic form: We mention that recently 3D Ising systems in oscillating magnetic fields were already considered by computer simulations [14]. However in [14] the authors study the hysteretic response of the system and no evidences for SR are discussed.
We will study the proposed system (1) from the viewpoint of SR both by computer simulations and by a theoretical approach.

Computer Simulation
To study the time evolution of the proposed system (1) first we considered a computer simulation with heath-bath dynamics. The time scale was chosen in a convenient way, setting the unit-time interval equal with the average characteristic time (τ ) necessary for the flip of a spin. We have taken this time interval τ as constant, and thus independent of the temperature. Although this assumption is just a working hypothesis we expect to give useful qualitative results. The spin flips were realized with the probabilities of the Metropolis [15] algorithm, choosing the spins randomly at each moment.
The simulations were performed on cubic lattices with W = N × N × N spins, considering the value of N up to 50. One simulation step was defined as W trials of changing spin orientations, and corresponds to a time interval τ . The period (P ) of the oscillating magnetic field will be also given in these τ units. The amplitude A of the magnetic field is considered already multiplied by µ B /k, and thus will have the dimensionality of the temperature (k is the Boltzmann constant). The temperature will be considered in arbitrary units, and the critical temperature of the infinite system (T c ≈ 4.444 J/k) will be set to 100 units. Starting the system from a random configuration we considered 10000 simulation steps to approach the dynamic equilibrium. The correlation function between the driving field B(t) and the magnetic response M(t) was studied after this during 10000 extra iterations. (The averaging in (3) is as a function of time, and t i = τ · i).
The correlation (σ) was studied as a function of: • the temperature (T), • the lattice size (N), • the amplitude of the magnetic field (A), and • the period of the oscillating magnetic field (P).

Theoretical approach
We propose now a mean-field like theory to describe the time evolution of the system.
Let us consider that P (M, t) dM is the probability for having the magnetization of the system between the values M and M + dM at a time moment t. We know from the classical fluctuation theories that for a system in thermal equilibrium with a heat-bath the probability P (x) of getting into a state x is proportional with the where: Up to second order terms one can consider: In this manner equation (5) will be written as with: A few remarks on the obtained approximative "master" equation ( • The equation will not necessary keep the normalization of P (M, t). For assuring that P (M, t) will have the meaning of probability, one must apply equation normalization condition at each time step. (W is the number of spins in the system.) • The value of C determines the speed of changing P (M, t) at each moment.
In reality C is temperature dependent, but as a first approximation one can consider it constant. Fixing it at a reasonable value we assume that will not influence a qualitative discussion on the phenomenon of SR, only the absolute values of the σ correlation.
The F (M, t) free energy of our system of W coupled S z = ±1 spins in a B(t) magnetic field can be computed using a mean-field approximation. We assume that with U the internal energy, T the temperature and S the entropy. For a cubic system, in the mean-field approximation we consider and: The first and second order derivatives of F (M, t) can be approximated numerically. Equation (8) with the condition (10) can be now solved also numerically for a given C value. Starting from a P (M, 0) initial distribution in principle we should be able to compute P (M, t). The σ correlation can be calculated as which is also computable.
Choosing the same units for J and B as in computer simulations we did solved numerically equation (8) with the normalization condition (10). B(t) was taken in the form given by (2) and we fixed J = 20. The "critical temperature" of the system without magnetic field became in this way T c ≈ 122. We also fixed W = 40 and considered different temperatures (T ) around T c , several periods (P) and several amplitudes (A) for the oscillating magnetic field. The constant C dt was taken as 0.01. We also checked that the overall picture of σ versus T is not significantly influenced by lowering the value of C dt.
Unfortunately the time scale of this theoretical approximation can not be obviously related to the time scale used in computer simulations. Comparision between theoretical and computer simulation results must be viewed under this assumption.

Results
Our simulation results and theoretical approximations are summarized in Figs. 1-6.
In Fig.1 we present a characteristic computer simulation result for the shape of the σ versus T curve. One will observe that in accordance with the predicted phenomenon of SR, at a given T r temperature σ presents a maximum. The tail of this resonance peak is nicely described by a power law (bottom picture). Analysing the scaling exponents for different simulations (different P , A and N values) we con- Figs. 3 and 4 presents the shape of the resonance peak for several values of the modulation amplitude A. Both theory (Fig. 4) and simulations (Fig. 3) predicts that the resonance temperature is almost independent of the modulation amplitude, exhibiting only very slight variation as a function of this (i.e. for higher amplitudes T r is shifted in the direction of smaller values). In contrast with this, the height of the peak depends sensitively on the values of the modulation amplitude.
As we concluded also in [13], for small lattices the T r resonance temperature is strongly dependent on the lattice size (N), and in the limit of relatively big lattices (N ≈ 20) is tending to a constant limiting value. In this sense for three different modulation periods P , the simulation results are presented in Fig. 5. We did not studied this dependence theoretically because our numerical calculations were technically limited by the values of W (i.e. only small lattices with W < 50 were computable). From Fig. 5 we also learn that the temperature T r is dependent on the modulation period P .
We studied the dependence of T r versus P both by simulations and theoretically.
The results in this sense are plotted in Fig. 6. Both simulations and theory predicts that for high periods the T r resonance temperature is tending to the T c "critical temperature" of the system. This dependence can be described by a

Conclusions
The first conclusion would be that both our computer simulations and theoretical data suggest that the phenomenon of SR should be detected when one studies finite ferromagnetic systems in oscillating magnetic fields. The characteristic peak of SR is obtained by studying the σ =|< B(t) · M(t) >| correlation as a function of temperature. For a given resonance temperature T r , this correlation σ exhibits a maximum.
Fixing the frequency, for small lattices (W < 4000) the T r resonance temperature is dependent on the lattice sizes and is tending to a limiting value for relatively large (W > 4000) lattices. The resonance temperature proved to be dependent also on the period of the magnetic field, and in the limit of large periods is converging exponentially to the critical temperature of the system. Because in real experimental conditions we are in the very high period limit (the time unit in simulations is set by τ , the characteristic time for the flip of a spin), we expect T r to be detected at T c .
We also concluded that the T r resonance temperature is not significantly influenced by the amplitude A of the oscillating magnetic field, the value A determining mainly the height of the resonance peak.
Our theoretical results proved to be in excellent concordance with the computer simulations, justifying thus the considered approximations.
We consider that from the viewpoint of magnetism an experimental study of the problem would also be of interest.

Acknowledgment
The work was done during a collaboration with the University of Bergen (Norway), supported by the Norvegian Research Council. I am grateful to L.P. Csernai for his continuous help.