![]() ![]() ![]() The problem now is to relate Equation ( 10) to the variables which are usually traced in computer simulations, like positions and momenta of all the particles. Where h ( x, t ) and ϕ ( x, t ) are, respectively, the energy density and the energy flux in the position x at time t. The fact that the defined flux is, on average, constant over all the particles in NESS is a loose indicator that it is a good candidate for being identified as the real physical heat flux. One ends up defining the energy flux particle-by-particle, thus implying the existence of a measurable quantity which is microscopic in nature. In numerical simulations, one monitors the dynamical evolution of individual particles, and the size of the system may be small compared to real physical systems. In fact, in real physical systems, the energy or the heat flux can be measured over mesoscopic regions of space. Even at the level of energy flux, there has been some concern as to what its precise definition should be, in function of the microscopic variables. The heat flux may be thought of as the energy flux, net of collective coherent motions. One related problem is defining the observable that can be identified with the heat flux. One-dimensional systems are characteristically anomalous models for heat conduction and transport. We show that the proposition can be explicitly constructed analytically by an ad hoc modification of the chosen model for the energy density. In an attempt to connect the dots, we propose that the locally integrated flux divided by the inter-particle distance is a good measure of the energy flux. On the other hand, it seems peculiar that this integral definition appears to capture a transport contribution, which may be called of convective nature, which is altogether missed by the former definition. We show that in stationary states, the first definition is resilient to several formally legitimate modifications of the (models of) energy density distribution, while the second is not. More worryingly, it does not satisfy in any obvious way an equation of continuity. ![]() It is, therefore, an integral quantity and not a local quantity. Conversely, the second definition corresponds to applying the introduced integral operator to the energy flux. By specializing to the case of chains with the particles’ order conserved, we show that the first definition corresponds to applying the differential continuity-equation operator after the application of the integral operator. We try to shed further light on their significance by introducing a novel integral operator, acting over movable boundaries represented by the neighboring particles’ positions, or some combinations thereof. The other relies somewhat elaborately on a definition of the flux, which is a function of every coordinate in the line. One definition equates the energy variation per particle to a discretized flux difference, which we here show it also corresponds to the flux of energy in the zero wavenumber limit in Fourier space, concurrently providing a general formula valid for all wavelengths. Example We’ve had an influx in clicks on our website thanks to the new ad we put out yesterday.We review two well-known definitions present in the literature, which are used to define the heat or energy flux in one dimensional chains. Similarly, to have an influx of something (the noun, as opposed to the verb of being in flux) is to experience a sudden growth or surge in it. Therefore the definition is very literal: something that is in flux (or in-flux, although usually, the hyphen is unnecessary) is flowing in and out, sometimes at a greater intensity and sometimes at a lesser one, and it never stays in one position or amount for long. So what is flux? The term comes from Latin, a form of the verb fluere, which means “to flow.” This linguistic origin evokes the idea of a river, constantly in motion and never staying the same. It can also be used in a number of different contexts, as we’ll explain below. Although English speakers have co-opted it into the vernacular, you might be surprised to learn that its origin does not even derive from English. In flux is an expression meaning fluid, not fixed, or given to change at any moment.
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