Dissipative work

This relation for the dissipated work is exact for a Gaussian work distribution of... 

Check. Use the Crooks relation (5.35) to check whether the forward and backward work distributions are consistent. Check for consistency of free energies obtained from different estimators. If the amount of dissipated work is large, caution may be necessary. If cumulant expressions are used, the work distributions should be nearly Gaussian, and the variances of the forward and backward perturbations should be of comparable size [as required by (5.35) for Gaussian work distributions]. Systematic errors from biased estimators should be taken into consideration. Statistical errors can be estimated, for instance, by performing a block analysis. 

Note that AA — xg > 0 in (6.36) is the average dissipated work of the NEW calculation in the complementary g) direction. Equation (6.36) shows that, given the same number of switching paths (/V), a simulation with a slower switch leads to a smaller variance in the free energy, since the value of A A — xg becomes smaller. Also note that AA — xg has its maximum value in the FEP case (instantaneous switch), and is zero when the switching path is reversible. 

The physical meaning of both entropies is now clear. Whereas Sp stands for the heat transferred by the system to the sources (Eq. (36)), the total dissipation term TS (Eq. (35)) is just the difference between the total mechanical work exerted on the system, W(r), and the reversible work, Wrev = AF. It is customary to define this quantity as the dissipated work, Waiss ... 

Figure 12. (Upper panel) Path entropy i(w) (Middle panel) path free-energy (w) = w — Ts(w), and (lower panel) Lagrange multipher X(w) equal to the inverse of the path temperature 1/7 (m ). is the most probable work value given by y(w P) = X,(rv P) = 0 or = 1 is the value of the work that has to be sampled to recover free energies from nonequilibrium work values using the JE. This is given by y(w() = l/T or d> (w() = 0 Wrev and Wdis are the reversible and average dissipated work, respectively. (From Ref. 117.)...

T. Speck and U. Seifert, Dissipated work in driven harmonic diffusive systems general solution and application to stretching rouse polymers. Eur. Phys. J. B 43, 521-527 (2005). 

Dissipated work work of plastic deformation hysteresis... 

The dissipated work potential is small as we mix two saturated steams at relatively low temperature levels. Mixing process 2 Available data from the steam tables are... 

In a steady-state mixing process, 50.25 kg/s of saturated steam (stream 1) at 501.15 K is mixed with 7.363 kg/s of saturated steam (stream 2) at 401.15 K. The mixer is well insulated and adiabatic. Determine the energy dissipation (work loss) if the surroundings are at 298.15 K. 

The work done consists of the reversible part and the dissipated work. We want to calculate these contributions individually with the help of an entropy balance. For the sake of simplicity we will only consider pure substances. According to Gibbs fundamental equation, we have... 

By definition, the mean pressure p = —1/3 5j rkk only includes normal stresses. In order to create a link between mean and thermodynamic pressure we will consider a cubic fluid element at a temperature T and of specific volume v, Fig. Al. We will now assume that the cube is at rest at time t = 0, so that the thermodynamic pressure p prevails inside the element. Now let us assume the mean pressure p is being exerted on the element from outside. When p > p the cube is compressed, should p < p then it expands. So, work — p dy is carried out by the external pressure p. This is equal to the work done during the volume change in the gas — p dy and the dissipated work. It therefore holds that dW = —p dV = —p dy + dWdi88 with the the dissipated work as... 

A number of sfraightforward applications of fhe Jarzynski relafion (offen termed "fast growth") soon followed [44,48-50]. If was quickly realized, however, that the poorly sampled tails of the work distribution in Eq. (7) contribute substantially to the free energy, posing difficulfies in convergence similar to those that occur with EXP [45,48,51]. Indeed, convergence was found fo require a number of simulations that increased exponentially in the typical dissipative work of the reverse process [45]. In practice, comparisons of fasf growfh mefhods and more standard equilibrium simulations seem to show that application of nonequilibrium measurements were often less efficient than equilibrium methods [39,52]. 

Because energy transferred to the system by dissipative work is converted to thermal energy, we could replace this work with an equal quantity of positive heat and produce the same overall change. The replacement of stirring work with heat was illustrated by experiment 3 on page 60. 

The shaft rotation angle d, which is the work coordinate for stirring work, is a property of the system but is not a state function, as we can see by the fact that the state of the system can be exactly the same for = 0 and = In. The work coordinate and work coefficient of work with a reversible limit are always state functions, whereas the work coordinate of any kind of dissipative work is not a state function. 

During a process with irreversible work, energy dissipation can be either partial or complete. Dissipative work, such as the stirring work and electrical heating described in previous sections, is irreversible work with complete energy dissipation. The final equilibrium state of an adiabatic process with dissipative work can also be reached by a path with positive heat and no work. This is a special case of the minimal work principle. 

Dissipative work is positive work with complete energy dissipation. In this type of work, the work coefficient vanishes in the limit of infinite slowness, and the work coordinate is not a state fimction. Examples are stirring work (Sec. 3.7.1) and the work of electrical heating (Sec. 3.8.2). 

Note that the work coordinate of any kind of dissipative work—work without a reversible limit—cannot appear in the expression for a total differential, because it is not a state function (Sec. 3.10). 

---