Rough equilibrium states

It follows from the above theorem that a rough system on the plane may possess only rough equilibrium states (nodes, foci and saddles) and rough limit cycles. As for separatrices of saddles, they either tend asymptotically to a node, a focus, or a limit cycle in forward or backward time, or leave the region G after a finite interval of time. 

The phase portraits for systems of dimension two and higher are illustrated in Figs. 11.2.4-11.2.7, respectively. Here, when l2 < 0, there are two rough equilibrium states a stable node and a saddle that approach each other as i2M... 

The parameter space here is partitioned into three regions Z o, D2 and i 4, joined at the origin. When /i G >4, the Eq, (11.2.27) has four rough equilibrium states, two of which are stable and two are unstable for /x >2, the equation has two rough equilibria, one stable and the other unstable for Do, there is no equilibrium states at all. 

Consider a one-parameter family of (r > 2) smooth dynamical systems in (n > 1). Suppose that when the parameter vanishes the system possesses a non-rough equilibrium state at the origin with one characteristic exponent equal to zero and the other n exponents lying to the left of the imaginary axis. We suppose also that the equilibrium state is a simple saddle-node, namely the first Lyapimov value I2 is not zero (see Sec. 11.2). Without loss of generality we assume /2 > 0. 

Summarizing, we can classify the rough equilibrium states in as follows ... 

Shilnikov, L. P. [1970] A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik 10, 91-102. 

This is concerned with the fact that in the case of the relaxation time, roughly speaking only half of all Brownian particles should leave the initial potential minimum to reach the equilibrium state, while for the profile of the decay time case all particles should leave the initial minimum. Expression (5.120), of course, is true only in the case of the sufficiently large potential barrier, separating the stable states of the bistable system, when the inverse probability current from the second minimum to the initial one may be neglected (see Ref. 33). 

What is the lowest possible energy for the harmonic oscillator defined in Eq. (5.10) Using classical mechanics, the answer is quite simple it is the equilibrium state with x 0, zero kinetic energy and potential energy E0. The quantum mechanical answer cannot be quite so simple because of the Heisenberg uncertainty principle, which says (roughly) that the position and momentum of a particle cannot both be known with arbitrary precision. Because the classical minimum energy state specifies both the momentum and position of the oscillator exactly (as zero), it is not a valid quantum... 

For liquid-vapor interfaces, the correlation length in the bulk is of t he order of atomic distance unless one is close to the critical point Hence the concept of local equilibrium is well justified in most practical circumstances For. solid surfaces above the roughening temperature, the concept also makes sense. Since the surface is rough adding (or removing) an atom to a particular part of the surface docs not disturb the local equilibrium state very much, and this sampling procedure can be used to determine the local chemical potential. This is the essence of the Gibbs-Thomson relation (1). 

For the kinetics of a reaction, it is critical to know the rough time to reach equilibrium. Often the term "mean reaction time," or "reaction timescale," or "relaxation timescale" is used. These terms all mean the same, the time it takes for the reactant concentration to change from the initial value to 1/e toward the final (equilibrium) value. For unidirectional reactions, half-life is often used to characterize the time to reach the final state, and it means the time for the reactant concentration to decrease to half of the initial value. For some reactions or processes, these times are short, meaning that the equilibrium state is easy to reach. Examples of rapid reactions include H2O + OH (timescale < 67 /is at... 

The Lotka model is an example of a rough system deviations of concentrations from their asymptotic values (2.1.24) occur independently on chosen parameters p, K, (3, i.e., small variations of these parameters cannot affect the way a system strives for the equilibrium state. 

A rough substrate with cavities can show hydrophobic behavior if air remains trapped in the cavities after a liquid drop is deposited on it. The effective surface energy of the substrate depends on the equilibrium state of the liquid-air interface inside the cavities. In general, multiple stable equilibrium states are possible, which... 

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