Linear regime

So far we have been considering the motion of a test polymer in an isotropic environment. We now consider a slightly different problem how does the orientational distribution function of polymers change under external fields such as a potential field / ( ) or a velocity gradient K. Let (ti t) be the probability that an arbitrarily chosen polymer is in the direction u. Since each polymer feels the external field as in eqn (8.15), the time evolution of W( t) can be described by  

An important point here is that in this problem, the environmental distribution function V, is the same as V( t) itself, and is now given 

Equations (9.36) and (9.37) give a nonlinear equation for W. The nonlinearity indicates a mean field character of the present theory it comes from identifying the distribution of the surrounding polymers with that of the test polymer. Equation (9.36) is thus different from the usual Smoluchowski equation, which is always linear in P. 

In the linear response regime, however, the nonlinearity of the kinetic equation is not important because there Dr can be replaced by Dr since the change in Dr appears only in the higher order perturbation. Therefore the linear response function is given by the same form as that in dilute solution except that Dr is much smaller than Z ,o- For example, consider a rodlike polymer which has permanent dipole moment p and isotropic polarizability (an = a = a ). The complex polarizability and the dynamic Kerr constant (per polymer) are given in the same form as eqns (8.94) and (8.104), 

The rotational diffusion constant can be obtained from these expressions. 

---

The fluorescence signal is linearly proportional to the fraction/of molecules excited. The absorption rate and the stimulated emission rate 1 2 are proportional to the laser power. In the limit of low laser power,/is proportional to the laser power, while this is no longer true at high powers 1 2 <42 j). Care must thus be taken in a laser fluorescence experiment to be sure that one is operating in the linear regime, or that proper account of saturation effects is taken, since transitions with different strengdis reach saturation at different laser powers. 

In the last section we considered tire mechanical behaviour of polymers in tire linear regime where tire response is proportional to tire applied stress or strain. This section deals witli tire nonlinear behaviour of polymers under large defonnation. Microscopically, tire transition into tire nonlinear regime is associated with a change of tire polymer stmcture under mechanical loading. 

This linear regime is achieved to within 90% of ] by 2/. For an ideal homogeneous membrane... 

It is seen that for a given (constant) gate voltage, the drain current first increases linearly with the drain voltage (linear regime), then gradually levels off to reach a saturation value (saturation regime). 

An alternative method to estimate the field-effect mobility consists of using the transconduclanee in the linear regime, given by Eq. (14.32). We noie that the... 

More recently, the Thiais group reported on temperature-dependent mobility of 6T and 8T down to 10 K [ 124]. In this case, the mobility was estimated from the linear regime and corrected for the contact resistance. Data for 8Tare shown in Figure 14-25. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

For simplicity, it is assumed that the equilibrium value of the macrostate is zero, x = 0. This means that henceforth x measures the departure of the macrostate from its equilibrium value. In the linear regime, (small fluctuations), the first entropy may be expanded about its equilibrium value, and to quadratic order it is... 

Evidently in the linear regime the probability is Gaussian, and the correlation matrix is therefore given by... 

As in the linear regime, consider the sequential transition X X2 —U X3. Again Markovian behavior is assumed and the second entropy is added separately for the two transitions. In view of the previous results, in the nonlinear regime the second entropy for this may be written... 

Now consider the transition E —> E in time x > 0. Most interest lies in the linear regime, and so for an isolated system the results of Section IID apply. From Eq. (50), the second entropy in the intermediate regime is... 

Figure 3 shows the profiles induced in a bulk system by an applied temperature gradient. These Monte Carlo results [ 1 ] were obtained using the static probability distribution, Eq. (246). Clearly, the induced temperature is equal to the applied temperature. Also, the slopes of the induced density and energy profiles can be obtained from the susceptibility, as one might expect since in the linear regime there is a direct correspondence between the slopes and the moments [1]. 

Figure 8 shows the r-dependent thermal conductivity for a Lennard-Jones fluid (p = 0.8, 7o = 2) [6]. The nonequilibrium Monte Carlo algorithm was used with a sufficiently small imposed temperature gradient to ensure that the simulations were in the linear regime, so that the steady-state averages were equivalent to fluctuation averages of an isolated system. 

De Koning, M., Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime. A variational approach, J. Chem. Phys. 2005, 122,... 

Fig. 3.10. Generic exponential curves of growth, normalized so that abscissa and ordinate are the same in the linear regime.

In our experiment, we first use R6G and LDS722 as the donor and acceptor. It is important to characterize the FRET signal of this pair in a linear regime below the lasing threshold to provide a performance reference for the OFRR FRET laser that we will investigate later. Figure 19.10 shows the characterization of their FRET behavior for varying acceptor/donor concentration ratios when the donor is excited with a low power CW laser at 532 nm. As shown in Fig. 19.10a, in the absence... 

---