The hydrodynamic equations for a gas of hard spheres with dissipative dynamics are derived from the Boltzmann equation. The heat and momentum fluxes are calculated to Navier-Stokes order and the transport coefficients are determined as explicit functions of the coefficient of restitution. The dispersion relations for the corresponding linearized equations are obtained and the stability of this linear description is discussed. This requires consideration of the linear Burnett contributions to the energy balance equation from the energy sink term. Finally, it is shown how these results can be imbedded in simpler kinetic model equations with the potential for analysis of more complex states.

The 'free energy principle' (FEP) has been suggested to provide a unified theory of the brain, integrating data and theory relating to action, perception, and learning. The theory and implementation of the FEP combines insights from Helmholtzian 'perception as inference', machine learning theory, and statistical thermodynamics. Here, we provide a detailed mathematical evaluation of a suggested biologically plausible implementation of the FEP that has been widely used to develop the theory. Our objectives are (i) to describe within a single article the mathematical structure of this implementation of the FEP; (ii) provide a simple but complete agent-based model utilising the FEP; (iii) disclose the assumption structure of this implementation of the FEP to help elucidate its significance for the brain sciences. be shown that minimising IFE makes the R-density a good approximation to posterior density of environmental variables given sensory data. Under this interpretation the surprisal term in the IFE becomes more akin to the negative of log model evidence defined in more standard implementations of variational Bayes [30].130 The Action-Perception CycleMinimising IFE by updating the R-density provides an upper-bound on surprisal but cannot minimise it directly. The FEP suggests that organisms also act on their environment to change sensory input, and thus minimise surprisal indirectly [1,2]. The mechanism underlying this process is formally symmet-135 ric to perceptual inference, i.e., rather than inferring the cause of sensory data an organism must infer actions that best make sensory data accord with an internal environmental model [9]. Thus, the mechanism is often referred to as

We present an analogy of Fano resonances in quantum interference to classical resonances in the harmonic oscillator system. It has a manifestation as a coupled behaviour of two effective oscillators associated with propagating and evanescent waves. We illustrate this point by considering a classical system of two coupled oscillators and interfering electron waves in a quasi-one-dimensional narrow constriction with a quantum dot. Our approach provides a novel insight into Fano resonance physics and provides a helpful view in teaching Fano resonances.

No abstract

The definition of the coherent phonon amplitude in Eq. ͑54͒ of our paper contains a minor misprint and should read D q ͑ t ͒ϵ͗ b q ͑ t ͒ϩb Ϫq † ͑ t ͒ ͘. ͑54͒ We would like to point out that the matrix element in Eq. ͑49͒ underestimates the piezoelectric electron-phonon interaction by a factor of 4 and should readThis same factor of 4 also appears in the driving function expressions in Eqs. ͑81͒, ͑85͒, and ͑86͒ which now read ͑86͒As a result, Figs. 10-17 in our paper need to be changed and the revised figures are shown below. The paragraph on page 235316-14 describing the results of Fig. 12 needs to be replaced by the following.''In our simulation, we find that piezoelectric and deformation potential contributions to the driving function are comparable. This is seen in Fig. 12 where S piezo (z) and S def (z), along with their sum, are plotted at tϭ2 ps. In this example, we find that S piezo (z) makes the dominant contribution to S(z,t) as can be seen in Fig. 12.'' FIG. 10. Driving function, S(z,t), for the coherent LA phonon wave equation as a function of position and time for the In x Ga 1Ϫx N diode structure and laser pumping parameters in Table II. S(z,t) is computed using the full microscopic expression of Eq. ͑79͒. FIG. 11. Driving function, S(z,t), in the simplified loaded string model for the coherent LA phonon wave equation as a function of position and time for the In x Ga 1Ϫx N diode structure and laser pumping parameters in Table II. FIG. 12. Driving function, S(z,t), in the simplified loaded string model at tϭ2 ps for the coherent LA phonon wave equation as a function of position for the In x Ga 1Ϫx N diode structure and laser pumping parameters in Table II. The total driving function, S(z,t), is the sum of piezoelectric and deformation potential contributions, S piezo (z,t) and S def (z,t).PHYSICAL REVIEW B 66, 079903͑E͒ ͑2002͒