What is the meaning of an entropy current?

Question

Imagine some kind of thermoionic device, where electrons jump off a metal in vacuum due to a fixed temperature bath and in the presence of a bias voltage.

In a steady state regime (and quasi-equlibrium regime), there will be a net particle current. Thanks to the first law of thermodynamics,

$$\Delta U=T\Delta S+\mu\Delta N$$, we can define the energy and entropy currents, $$\dot{U}=T\dot{S}+\mu J$$, where $U$ is the energy, $S$ is the entropy, $T$ the temperature, $N$ the number of particles, $\mu $ the chemical potential of the metal, $\dot{U}$ the energy current, $\dot{S}$ the entropy current and $J$ the particle current.

In a microscopic picture, $J\propto \int f(E)\tau(E) \mathrm{d}E$ and $\dot U\propto \int E f(E)\tau(E) \mathrm{d}E$, where $\tau(E)$ takes into account the probability of escaping the metal and $f$ is the Fermi-Dirac distribution.

By construction, $\dot{U}$ will be positive (as long as the metal is not receiving electrons). But $T\dot{S}=\dot{U}-\mu J$ can be negative or positive.

What is the physical interpretation of $T\dot{S}$? Why it can take positive and negative values?

Answer 1

we can define the energy and entropy currents

What you defined are the rates of change ($\dot U$, $\dot S$). In the case of energy, which is conserved, the rate of change must correspond to the net flow. But entropy is different; it is not conserved.

Your example can be generalized to

$$T\dot S=\dot U-\sum_i F_i\frac{\partial x_i}{\partial t},$$

where $\dot S$ is the rate of system entropy increase and $F_j$ and $x_j$ are respectively generalized forces and displacements; examples include

• negative pressure and volume
• surface tension and surface area
• electric field and polarization
• voltage and charge
• magnetic field and magnetic moment
• force and distance
• stress and volumetric strain
• chemical potential and amount of matter

Along with energy, the generalized displacements above are conserved except for the electrical polarization and magnetic moment; thus, we ignore those forms of work for simplicity. By the divergence theorem expressed as $\frac{dA}{dt}=-\nabla\cdot\boldsymbol{J_A}+\dot\sigma =-\nabla\cdot\boldsymbol{J_A}$, where $\boldsymbol{J_A}$ is the flux of A and $\dot\sigma$ represents generation (absent for a conserved quantity), we obtain the continuity relation

$$T\dot S=-\nabla\cdot\boldsymbol{J_U}+\sum_iF_i\nabla\cdot\boldsymbol{J_i}.$$

From the chain rule, $A\nabla\cdot\boldsymbol{B}=-\boldsymbol{B}\nabla\cdot A+\nabla\cdot (A\boldsymbol{B})$; thus, letting $A=\frac{1}{T}$,

\begin{align}T\dot S&=T\left(\boldsymbol{J_U}\cdot\nabla\frac{1}{T}-\nabla\cdot\frac{\boldsymbol{J_U}}{T}-\sum_i\boldsymbol{J_i}\cdot\frac{F_i}{T}+\nabla\cdot\frac{\sum_iF_i\boldsymbol{J_i}}{T}\right)\\&=T\left[\underbrace{\boldsymbol{J_U}\cdot\nabla\frac{1}{T}-\sum_i\boldsymbol{J_i}\cdot\frac{F_i}{T}}_{\substack{\text{this looks like}\\\text{a generation term}}}\underbrace{-\nabla\cdot\left(\frac{\boldsymbol{J_U}-\sum_iF_i\boldsymbol{J_i}}{T}\right)}_{\substack{\text{this looks like}\\\text{a flux term}}}\right].\end{align}

Looking again at the divergence theorem, we infer a relation

$$T\dot S=T(\dot \sigma-\nabla\cdot \boldsymbol{J_S}),$$

where we identify

$$\dot\sigma =\boldsymbol{J_U}\cdot\nabla\frac{1}{T}-\sum_i\boldsymbol{J_i}\cdot\nabla\frac{F_i}{T}$$

as the entropy generation rate and $$\boldsymbol{J_S}=\frac{\boldsymbol{J_U}-\sum_iF_i\boldsymbol{J_i}}{T}$$

as the conserved entropy flux or current.

As derived here, for example, $\boldsymbol{J_U}=\boldsymbol{J_Q}+\boldsymbol{J_W}+\boldsymbol{J_H}$, where $\boldsymbol{J_Q}$ is the heat flux, $\boldsymbol{J_W}$ is the work flux (excluding flow work) and $\boldsymbol{J_H}$ is the enthalpy flux related to mass transfer (recall that the molar enthalpy $h=\mu+Ts$, where $\mu$ is the chemical potential). The subtraction of $\sum_iF_i\boldsymbol{J_i}$ from $\boldsymbol{J_U}$ above effectively removes $\boldsymbol{J_W}$ and $\sum_i\mu_i\boldsymbol{J_i}$ for mass flow $\boldsymbol{J_i}$. This allows us ultimately to write the above relation as

$$\boldsymbol{J_S}=\frac{\boldsymbol{J_Q}+T\sum_is_i\boldsymbol{J_i}}{T},$$

where $s_i$ is the entropy, if any, carried by generalized displacement $i$ (e.g., the flow of charge carriers).

In addition, applying the chain rule to the entropy generation rate gives

$$\dot\sigma=\frac{1}{T^2}\left(-\boldsymbol{J_U}\cdot\nabla T+\sum_iF_i\boldsymbol{J_i}\cdot \nabla T-T\sum_i\boldsymbol{J_i}\cdot\nabla F_i\right);$$ thus, $$T\dot\sigma=\underbrace{-\boldsymbol{J_S}\cdot\nabla T}_{\substack{\text{contributes to entropy}\\\text{generation from flow down}\\\text{a temperature gradient}}}+\underbrace{\sum_i-\boldsymbol{J_i}\cdot\nabla F_i.}_{\substack{\text{contributes to entropy}\\\text{generation from flow down}\\\text{all other gradients}}}$$

What this tells us is that entropy can be transferred in and out of a system through heating and through the entropy carried by matter, for example. This is $\boldsymbol{J_S}$.

In addition, entropy is generated whenever flow of a generalized displacement (all $\boldsymbol{J_i}$ plus the entropy flow $\boldsymbol{J_S}$ itself) is driven by a gradient in any internal property (all $\boldsymbol{\nabla F_i}$ plus temperature $\boldsymbol{\nabla T}$). The direction of spontaneous flow is opposite the direction of the gradient in the intensive property; in this way, Nature smooths out disturbances. This is $\boldsymbol{\dot\sigma}$.

Depending on how much entropy you transfer in and out and how much entropy you generate, $\boldsymbol{\dot S}$ can be positive or negative.