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In general, the definition that a dichotomous noise is symmetric or not is made as follows. First, we assume that η (t) is a general dichotomous noise which takes two values u and w ( u and w can be zero, positive or negative, the transition rate from u to w is λ1 and vice versa is λ2). Then, if η (t) =0, we can get u λ2 +w λ1 =0, i.e., -u/w= λ1 / λ2. Now, if -u/w=1, we say that the dichotomous noise η (t) is symmetric; otherwise, we say that it is asymmetric. Here, we must notice that if η (t) even if -u/w=1, we can not say that η (t) is symmetric.
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The relationship between the stationary probability current J and stationary average velocity V s = dx/dt V s for Eq. is s = (d-c ) J (see Ref.).
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The relationship between the stationary probability current J and stationary average velocity V s = dx/dt V s for Eq. is s = (d-c ) J (see Ref.).
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For a colored Gaussian noise, one can note (without a proof) that, usually, the results (or properties) of stochastic particles subject to a this noise (Gaussian colored) are the same as the ones for an equivalent system driven by a symmetric dichotomous noise.
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For a colored Gaussian noise, one can note (without a proof) that, usually, the results (or properties) of stochastic particles subject to a this noise (Gaussian colored) are the same as the ones for an equivalent system driven by a symmetric dichotomous noise.
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If we add a thermal (Gaussian white) noise in Eq. , two methods can be used for us to solve the problem. One is the numerical simulation method directly using the Langevin equation, but this will result in a long-time calculation for the workstations or computers and, of course, we cannot get the exact (or approximate) analytical formula for the stationary probability current. The other is the method using the method in Ref. together with the mean-field approximation in Ref. but, for this method, the exact (or approximate) analytical formula for the stationary probability current cannot be gotten because of the complexity caused by the coupling.
-
If we add a thermal (Gaussian white) noise in Eq., two methods can be used for us to solve the problem. One is the numerical simulation method directly using the Langevin equation, but this will result in a long-time calculation for the workstations or computers and, of course, we cannot get the exact (or approximate) analytical formula for the stationary probability current. The other is the method using the method in Ref. together with the mean-field approximation in Ref. but, for this method, the exact (or approximate) analytical formula for the stationary probability current cannot be gotten because of the complexity caused by the coupling.
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