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Extra resources for Global Theory of Dynamical Systems
Example text
5) where Kl =T -GN O and K2 =Ga. Here K2 is always positive. 2) of the previous example, the difference being that now,our particle is restrained by a restoring force, which instead of a cubic nonlinearity has a quadratic one. 7, and the only stable steady state is x =0. , the number of coherent photons, see below) in the cavity is zero. 8, so there exists a stable steady state with x *0, x* = [K 1 [/K 2, where the device behaves as a coherent source of radiation. 3 and 4. Originally a "coherent wave" means a wave of the type exp[J(wt - 15) i' If all complexions are a priori equiprobable, then the probability of having one of them responsible for the given macrostate is P =W- 1. Question: What is the uncertainty-and the information to be deduced there from-about which complexion is responsible for the observed macros tate? Think of the system under consideration as divided into two parts 1 and 2. Let WI and W2 be the number of complexions responsible for the state of the first half and the second half respectively; of course, W=W 1W2. 38 xln 2 x10- 23 J/K. 7]. We can now elaborate the conditions under which the entropy of a system increases irreversibly with time, or examine which are the premises under which a disturbed system left alone goes with probability one to a state of maximum disorder or perfect symmetry. Moreover, as we shall see, this state, where all dyadic velocity cross-correlations behave as delta functions, is stable. lnP. 27) where P. /N). So, for N,. large, we have the equivalent expression , N~= ' Ni AN. A Ni (In N.