Classical Nernst engine

We introduce a simple model for an engine based on the Nernst effect. In the presence of a magnetic field, a vertical heat current can drive a horizontal particle current against a chemical potential. For a microscopic model invoking classical particle trajectories subject to the Lorentz force, we prove a universal bound $3-2\sqrt{2}\simeq 0.172$ for the ratio between the maximum efficiency and the Carnot efficiency. This bound, as the slightly lower one $1/6$ for efficiency at maximum power, can indeed be saturated for a large magnetic field and small fugacity.

Figure 1

Scheme of the classical Nernst engine. The vertical heat current (red arrow) between reservoir $C_3$ and $C_1$ with $T_3>T_1$ drives a horizontal particle current (grey arrow) from reservoir $C_4$ to $C_2$ with ${\mu }_2>{\mu }_4$. The circular arrows show typical trajectories for a strong magnetic field $\mathbf{B}$. Their radius and number reflect the temperature and chemical potential, i.e., density, of the respective reservoir they originate from. The coordinate $0\le s\le 2\pi R$ parametrizes the boundary.

Figure 2

Geometry of a single trajectory leaving the reservoir $C_i$ at the position $s_{\text{in}}$ with an angle $\vartheta >0$ and entering the reservoir $C_{i+1}$ at $s_{\text{in}}+\mathrm{\Delta }s$. The boundary appears as a straight line, since the Larmor radii are typically small compared to the radius of the central region in the strong field regime. For further symbols, see main text.