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# 物理代写|热力学代写Thermodynamics代考|PHY360 Selective Decay

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## 物理代写|热力学代写Thermodynamics代考|Selective Decay

This rather frustrating series of failures may suggest that the problem lies in the common assumption of all the proposals discussed so far: LTE. For example, the equations of motion in many physical systems show that different quantities evolve with different time-scales and spatial scales; this result does not rely on LTE (explicitly, at least). Accordingly, it is at least conceivable that relaxation satisfies the ‘selective decay’ scenario.

The selective decay hypothesis is characterized by the following. If one considers the ‘ideal invariants’ of the system (namely, the quantities which would remain exactly constant during the evolution of the system should no dissipation occur), once dissipation has been introduced these quantities do not remain constant but start decaying, unless the external world somehow maintains their value constant. It is often found that one of these quantities is somehow ‘better conserved’ or ‘more rugged’ than others, i.e. that its typical decay time is much longer than the decay time of other quantities. ${ }^{11}$ If one minimizes the expression for the poorly conserved invariant subject to the constraint that the rugged invariant is conserved using the technique of Lagrange multipliers (Sect. A.3), an Euler-Lagrange equation for the field variables in the relaxed state results. The Lagrange multiplier is the ratio of the poorly conserved invariant to the ruggedly conserved one.

Typically, selective decay applies to problems in two-dimensional and threedimensional magnetohydrodynamics (‘MHD’), where the couples ‘rugged invariant versus poorly conserved invariant’ are ‘energy versus mean square vector potential’ and ‘energy versus magnetic helicity’, respectively. In MHD, for example, a well-known example of relaxed state is described by Taylor’s principle of minimum magnetic energy $\propto \int|\mathbf{B}|^2 d \mathbf{x}$ with fixed magnetic helicity $\int(\mathbf{A} \cdot \mathbf{B}) d \mathbf{x}$ (where $\mathbf{B}=\nabla \wedge \mathbf{A}$ and $\mathbf{A}$ is the vector potential) [25]. In Hall MHD, i.e. a macroscopic description of magnetized plasmas (made of two species, electrons and ions with ion mass $m_{i o n}$ and ion electric charge $q_{i o n}$ ) where electrons are effectively decoupled from ions, the couple ‘rugged invariant versus poorly conserved invariant’ is ‘total (magnetic + kinetic) energy’ versus ‘magnetic helicity and generalized ion helicity $\int(\mathbf{V} \cdot \boldsymbol{\Omega}) d \mathbf{x}$ ‘, with $\mathbf{V} \equiv \mathbf{v}+\frac{q_{\text {ion }}}{m_{\text {ion }}} \mathbf{A}$ and $\boldsymbol{\Omega} \equiv \nabla \wedge \mathbf{V}$. Taylor’s principle is replaced by Turner’s principle [26] of minimization of total energy with two constraints: fixed magnetic helicity and fixed generalized helicity. Remarkably, and in qualitative agreement with Kirchhoff’s principle of Sect. 5.3.1, in order to describe plasmas in the solar corona it has been postulated [27] to replace Turner’s principle with the constrained minimization of Joule heating power; fixed generalized ion helicity and its electron counterpart are the constraints. ${ }^{12}$

## 物理代写|热力学代写Thermodynamics代考|Maximal Entropy

The principle of ‘maximal entropy’13 dictates that the air in a room initially distributed in clumps moves towards smooth uniformity; thermodynamic equilibrium does not admit large-scale structures. However, for a system with a constrained phase space, maximal entropy can generate large-scale structures as a long-lived intermediate state. Remarkably, no LTE is explicitly invoked. To apply the principle of maximal entropy, one needs to consider a discrete or quantized version of the field variables. If we have $N$ such quanta of the field, ${ }^{14}$ we consider the number of ways these $N$ quanta can be arranged in a given state (like spins up or down). The most probable state is the one with the most permutations or the highest entropy subject to other constraints (such as conservation of energy and particle number); here entropy is defined as the logarithm of the number of permutations times Boltzmann constant. The description of the system is perfectly analogous to the familiar description of the $2 \mathrm{D}$ spin system in statistical mechanics of thermodynamic equilibrium [32]. The maximal entropy perspective addresses the question: are these observed large-scale, self-organized structures in some sense statistically more probable than other less simple ones?

Again, our room is a thermodynamically open system. The room exchanges either heat (across the closed window) or both heat and mass (across the open window) with the external world. In a relaxed state, the long-lived, large-scale structures are supposed to live not just for a long time, but indefinitely. The relevance of these approaches to the relaxed state of our room is, therefore, yet to be proven.

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