Collisional invariants for the phonon Boltzmann equation

For the phonon Boltzmann equation with only pair collisions we characterize the set of all collisional invariants under some mild conditions on the dispersion relation

Approach to equilibrium for the phonon Boltzmann equation

We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities.Comment: 23 page

Phosphorus and nitrogen cycling in forest soils depending on long-term nitrogen inputs

Foliar phosphorus (P) contents have been decreasing in a range of temperate forests in Europe and North America during the last decades, and one reason for this might be atmospheric nitrogen (N) deposition (1,2,3). Therefore, we studied the effect of N inputs on P and N cycling in long-term N fertilization experiments in temperate forests. The aim of the study was to test how increased N inputs affect P and N cycling in forest soils. We sampled the organic layer of three N fertilization experiments in the USA (Harvard Forest, Cary Institute and Bear Brook), that are between 17 and 25 years old. Net N and P mineralization rates were determined along with microbial biomass, enzyme activities and soil C, N and P stoichiometry. Total C and N concentrations in the organic layer (Oe+Oa horizon) increased significantly due to long-term fertilization in Harvard Forest and the same trend was observed in the two other experiments that are based on lower N fertilization rates. Contrariwise, total P concentrations in the organic layer decreased on average by 15% due to N fertilization, while C:P ratios increased by 60%. Phosphatase activity was elevated in the N fertilized soils in all experiments by a factor of 2 to 5, and the ratio of chitinase:phosphatase activity was on average decreased by 30%, indicating that specifically phosphatase production was upregulated. The results imply that trees and/or microorganisms invested more N in the production of phosphatases in the N fertilized soils than in the non-fertilized controls. Net P mineralization did not change consistently with N inputs, indicating that mineralized P was quickly taken up by the plants in most of the N fertilized soils. In contrast, net N mineralization increased in all experiments in response to N fertilization, while microbial biomass C was only little affected by N fertilization In conclusion, the experiments indicate that high inputs of N in temperate forest ecosystems lead to increased P demand and hence to increased phosphatase activity. Moreover, the decreased P concentration and the elevated C:P ratio of the organic layer indicate that P is preferentially mineralized and taken up by plants. Our results support the hypothesis that increased atmospheric N inputs are the reason for an emerging P limitation in temperate forests

Complementarity relation for irreversible process derived from stochastic energetics

When the process of a system in contact with a heat bath is described by classical Langevin equation, the method of stochastic energetics [K. Sekimoto, J. Phys. Soc. Jpn. vol. 66 (1997) p.1234] enables to derive the form of Helmholtz free energy and the dissipation function of the system. We prove that the irreversible heat Q_irr and the time lapse $Delta t} of an isothermal process obey the complementarity relation, Q_irr {Delta t} >= k_B T S_min, where S_min depends on the initial and the final values of the control parameters, but it does not depend on the pathway between these values.Comment: 3 pages. LaTeX with 6 style macro

A hyperbolic conservation law and particle systems

In these notes we consider two particle systems: the totally asymmetric simple exclusion process and the totally asymmetric zero-range process. We introduce the notion of hydrodynamic limit and describe the partial differential equation that governs the evolution of the conserved quantity – the density of particles p(t,.). This equation is a hyperbolic conservation law of type ətp(p, u) + vF(p(t, u)) = 0, where the flux F is a concave function. Taking these systems evolving on the Euler time scale tN, a central limit theorem for the empirical measure holds and the temporal evolution of the limit density field is deterministic. By taking the system in a reference frame with constant velocity, the limit density field does not evolve in time. In order to have a non-trivial limit, time needs to be speeded up and for time scales smaller than tN 4=3, there is still no temporal evolution. As a consequence, the current across a characteristic vanishes up to this longer time scale.Fundação para a Ciência e a Tecnologia (FCT) - bolsa SFRH/BPD/39991/2007Fundação Calouste Gulbenkian - projecto "Hydrodynamic limit of particle systems

Clausius inequality and optimality of quasi static transformations for nonequilibrium stationary states

Nonequilibrium stationary states of thermodynamic systems dissipate a positive amount of energy per unit of time. If we consider transformations of such states that are realized by letting the driving depend on time, the amount of energy dissipated in an unbounded time window becomes then infinite. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work performed along any given transformation. We then show that the renormalized work satisfies a Clausius inequality and prove that equality is achieved for very slow transformations, that is in the quasi static limit. We finally connect the renormalized work to the quasi potential of the macroscopic fluctuation theory, that gives the probability of fluctuations in the stationary nonequilibrium ensemble

How large are present-day heat flux variations across the surface of Mars?

©2016. American Geophysical UnionThe first in situ Martian heat flux measurement to be carried out by the InSight Discovery‐class mission will provide an important baseline to constrain the present‐day heat budget of the planet and, in turn, the thermochemical evolution of its interior. In this study, we estimate the magnitude of surface heat flux heterogeneities in order to assess how the heat flux at the InSight landing site relates to the average heat flux of Mars. To this end, we model the thermal evolution of Mars in a 3‐D spherical geometry and investigate the resulting surface spatial variations of heat flux at the present day. Our models assume a fixed crust with a variable thickness as inferred from gravity and topography data and with radiogenic heat sources as obtained from gamma ray measurements of the surface. We test several mantle parameters and show that the present‐day surface heat flux pattern is dominated by the imposed crustal structure. The largest surface heat flux peak‐to peak variations lie between 17.2 and 49.9 mW m−2, with the highest values being associated with the occurrence of prominent mantle plumes. However, strong spatial variations introduced by such plumes remain narrowly confined to a few geographical regions and are unlikely to bias the InSight heat flux measurement. We estimated that the average surface heat flux varies between 23.2 and 27.3 mW m−2, while at the InSight location it lies between 18.8 and 24.2 mW m−2. In most models, elastic lithosphere thickness values exceed 250 km at the north pole, while the south pole values lie well above 110 km