But in non-equilibrium cooling process the first solid formed may have a higher solute composition than the next solid that forms leading to the formation of a cored structure. Equilibrium cooling gives us a desired microstructure and is the process normally used in industries to make metal bars from the molten state. Sufficient time is available for the solute to diffuse in non-equilibrium cooling, but in equilibrium cooling there is no diffusion of solute.
In equilibrium cooling, the cooling rate is so slow, whereas, non-equilibrium cooling process has a higher cooling rate. The typical solidification or cooling process in industries is the non-equilibrium cooling process. But in non- equilibrium cooling process the first solid formed may have a higher solute composition than the next solid that forms leading to the formation of a cored structure.
Equilibrium cooling gives us a desired microstructure and is the process Ask your question! Help us make our solutions better Rate this solution on a scale of below We want to correct this solution. Tell us more Hide this section if you want to rate later. Questions Courses. It leaves plenty of room for different behaviors which may allow or not to use the formalism of equilibrium thermodynamics.
Condition n. However, there is the possibility that a system of this class could be treated as a system in Local Thermodynamic Equilibrium LTE. The condition is that each subvolume of the system containing a macroscopic number of degrees of freedom could be considered at equilibrium over the typical scale of time connected to the macroscopic motion.
For instance, a steady jet of gas coming out from a high temperature source, could be in LTE conditions if the velocity distribution of small regions of the jet, in the center of mass frame, would be well approximated by a Maxwellian. In that case, one can speak about local density, temperature, internal energy, etc. Equilibrium impies that the thermodynamic coordinates and hence their functions do not change in time.
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Nonequilibrium temperature versus local-equilibrium temperature J. Jou Phys. Next, we impose the constraint of detailed balance,. Substituting equation 24 into 20 gives:. In other words, under the assumption of detailed balance P x, t uniquely determines J x, t , and then the two together determine the potential U x, t , by equation The next challenge is to choose a periodic P ps x, t such that i its time average is equal to P ss x equation 19 b , and ii the corresponding time-averaged current is equal to J ss equation 19 a.
An explicit construction with these properties is shown in the next subsection. We obtain:. Therefore, to satisfy equations 19 a and 19 b , we can choose a probability distribution P ps x, t with the desired temporal average and with a non-vanishing averaged current, and then match the averaged current by the rescaling of T and its sign by time reversal. Lastly, given P ps x, t and J ps x, t , we can use equation 23 to construct U x, t.
Importantly, the construction above has a lot of freedom: the only constraints on P ps x, s are its time average, positivity, smoothness and a non-vanishing average current. This freedom implies that there exist many periodic potentials generating the same time-averaged current and probability distribution. We now illustrate this procedure with a simple example. Let us construct U x, t that drives a time-averaged current and probability distribution. As discussed above, P ps x, s can be chosen arbitrarily, provided it is positive, normalized and has the correct time average and non-vanishing current.
The specific choice. Equation 24 implies in this case. Figure 1. The example in equation Upper panel: P x, s , given in equation Middle panel: the corresponding J x, s for which the driving is assured to be detailed balance.
Lower panel: the corresponding driving U x, u , given by equation As we have seen, there is considerable freedom in constructing a protocol U x, t that drives a target and. Moreover, in section 3 it was shown that the entropy production rate of a SP always exceeds that of a NESS, when both share the same time averaged probability distribution and current; see equation It is therefore natural to look for the protocol U x, t that drives the target averages at the minimal entropy production cost.
In other words, we would like to solve the following minimization problem:. Solving the optimization problem directly is challenging, but unnecessary: it is possible to construct a specific protocol that asymptotically approaches the bound in equation The construction of this protocol for generic is given in the appendix. In this section a simple example of this construction is demonstrated. Let us consider driving a given current, , with.
In other words, we consider a probability distribution with a fixed shape that moves at a constant velocity u. In this case, the continuity condition in equation 24 implies that.
The target current is set to be , which gives us. Substituting the above results into equation 6 , we get after some trivial algebra:. We see that is minimal when is maximal. Comparing with equation 18 we see that, in this limit, the entropy production of the periodically driven state approaches the bound set by the corresponding steady state value.
In this work we have discussed similarities and differences between two types of driving that maintain a diffusive system on a ring out of equilibrium: periodic variations of a potential along the ring, and static driving by breaking the detailed balance condition.
We have shown that the two scenarios can drive any averaged current and probability distribution, but in contrast to discrete state Markovian systems there is no full control in terms of the averaged entropy production.
Moreover, it was shown that the averaged entropy production of a steady state driving is smaller than that of a system driven by periodic changes in the potential that achieves the same averaged current and probability distribution. In terms of applications, this implies that the common driving in biological molecular motors—burning fuel and reaching a steady state—has a lower thermodynamic cost, i.
This result is different than what was obtained in a coarse-grained description of the same system—discrete state Markovian modeling—since a diffusive description reduces the number of controllable parameters i. Many important aspects were not discussed in this work and they could be subjects to future investigations. We thank R Zia for pointing out the physical interpretation of. In section 6 we analyzed a specific example where the entropy production rate of a SP can get arbitrarily close to that of a NESS with the same time averaged current and probability distribution.
In this appendix we generalize this construction for arbitrary target and. We first show that by appropriately choosing x 0 t we can construct the target time averaged probability:. By controlling the speed at which this delta function moves across each point we can manipulate the time averaged probability at this point. Specifically, for this example equation A. More generally, applying the convolution theorem of Fourier transforms to equation A. Note that the above equation shows that not any f x can serve for our construction—for example, if the right hand side of the above equation vanishes for some x then the corresponding diverges.
From the function , we construct , and then invert t x 0 to obtain x 0 t. Next, let us consider the current.
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