Forced stationarity

In regards to the stationarity of the reaction-diffusion process, it should be emphasised that the number of the B atoms diffusing across the ApBq layer is always equal to their number combined by the A surface into the ApBq compound at interface 1, if the growth of this layer is not accompanied by the formation of other compounds or solid solutions. The case under consideration is characterised by a kind of forced stationarity due to (z) the impossibility of any build-up of atoms at interfaces between the solids, (z z) the limited number of diffusion paths in the ApBq layer for the B atoms to travel from interface 2 to interface 1 and (Hi) the finite value of the reactivity of the A surface towards the B atoms. The stationarity is only... 

Balance of Forces It is convenient to define the L function L(x,Xy) =f(x) + g(xfX + h(xfv, along with "weights or multipliers X and v for the constraints. The stationarity condition (balance of forces acting on the ball) is then given by... 

A comparison between the microslip observed for PET-PET contacts at sliding velocities of 29 and 145 yms is made in Table II. The b value at 145 pms is considerably greater than that at 29 pms despite the much higher normal force at the lower velocity which, as indicated above, would tend to produce greater non-stationarity. The e values for sliding at 145 pms l were also greater than those at 29 pms at comparable loads except at very small loads when the microslip was similar ( ). 

The measurements as a function of increasing normal load demonstrate that there is not a simple relationship between microslip and non-stationarity. Greater damage is produced during junction breakdown at high loads but the microslip decreases with normal load. It is believed that this reduction in microslip with load is related to the corresponding increase in the frictional force with load ( ). 

There seems to be a slightly formal relation between these two principles when the principle of least dissipation of energy is expressed in the form of Eq. 60. If the linear relations are used, the dissipation function (p J, J) is numerically equal to the local entropy production and hence the minimum value of Eq. 152 agrees with the maximum of Eq. 59 compatible with the conditions 57 and 58. However, in the case of Eq. 60, the / s are varied under the conditions 57 and 58., in contrast with Eq. 152 for which the forces are varied and which gives the stationarity conditions 57 and 58, being identical with Eqs. 148 and 149, respectively. Both of these principles determine the spatial... 

If the flows and the forces do not satisfy the above condition, it is impossible to find the function which decreases monotonically and which agrees with the entropy production for the stationary state. In this case the condition for the stationarity is written, according to Eqs. 141 and 161, as... 

The statistical properties of the random force f(0 are modeled with an extreme economy of assumptions f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean (f(0 = 0), uncorrelated with the initial value v(t = 0) of the velocity fluctuations, and delta-correlated with itself, f(0f(t ) = f25(t -1 ) (i.e it is a purely random, or white, noise). The stationarity condition is in reality equivalent to the fluctuation-dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes the value of y. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression y(f)v(t) = exp [v(0)v(0) -+ ylM °, where Xg = In equilibrium, the long-time asymptotic value y/M must coincide with the equilibrium average (vv) = (k TIM)t given by the equipartition theorem (with I being the 3 X 3 Cartesian unit tensor), and this fixes the value of y to y=... 

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