Exact four-vector work distribution and covariant Jarzynski's equality for a relativistic particle in an expanding piston

Abstract

We investigate the non-equilibrium four-vector work in an expanding relativistic piston. By deriving the exact work distribution in this pedagogical model, we verify the covariant form of Jarzynski's equality. We find that the joint distribution of four-vector work $(W^0, W^1)$ concentrates on the origin and some curves in the $(W^0, W^1)$ space, rather than being smoothly distributed. In the non-relativistic limit, our model consistently recovers the non-relativistic dynamics. We further demonstrate that the momentum component of four-vector work remains significant in both the Lorentz-relativistic and Galilean-relativistic frameworks. In addition, we introduce a novel geometrical technique for analyzing the dynamics of relativistic collision processes, which can be straightforwardly extended to three-dimensional piston models.

Figures (11)

• Figure 1: A relativistic particle moving in a one-dimensional cylinder. The left boundary is fixed in the laboratory reference frame, while the right boundary is a elastic piston with constant velocity $v_p$. An inertial observer is moving relative to the laboratory frame with a constant velocity $-u$. The initial position and velocity of the particle are $x_i$ and $v_i$, respectively.
• Figure 2: Worldline of a particle. (Left) The dashed line denotes the worldline of a particle. Line $P$ is the worldline of the piston. In the cylinder's frame, the worldline of the cylinder (Line $C$) coincides with the time axis ($t$-axis). An initial state $i$ lies on the $x$-axis, which is a space-like hypersurface ($\Sigma_{\mathrm{ini}}$) orthogonal to worldline $C$. The corresponding final state $f$ also lies on a space-like hypersurface ($\Sigma_{\mathrm{fin}}$, which represents the isochronous surface $t=\tau$ in the cylinder's frame). (Right) In an arbitrary inertial observer's frame, the initial states no longer belong to any isochronous surface. However, the relation that $\Sigma_{\mathrm{ini}}/\Sigma_{\mathrm{fin}}$ being orthogonal to the line $C$ is still valid.
• Figure 3: Orthogonal reflection and auxiliary worldline. This figure demonstrates the original worldline segments and the single straight auxiliary worldline after multiple orthogonal reflections. It also shows the results of the reflections of $C_k$'s and $P_k$'s. Meanwhile, the hyperbolas $\Sigma_{\mathrm{C}}$, $\Sigma_{\mathrm{P}}$ denote the events that shares the same space-time interval relative to $O$. The intersection between $C_k$ and $\Sigma_{\mathrm{C}}$ is $C_{\mathrm{f,k}}$, and the intersection between $P_k$ and $\Sigma_{\mathrm{P}}$ is $P_{\mathrm{f,k}}$, which are denoted by the red and blue dots. In this figure, the velocity $-u$ of the observer is nonzero, therefore, $C_0$ does not coincide with the $t$-axis. For the same reason, the $\Sigma_{\mathrm{ini}}$ is no longer isochronous.
• Figure 4: The "shadow" of $\Sigma_{\mathrm{fin,n}}$ on $\Sigma_{\mathrm{ini}}$ is the possible initial position for a particle with initial velocity $v_i$ to collide $n$ times. The "parallel light" is represented by the green lines, and the slope of the "parallel light" is determined by $v_i$. In this example, the boundaries of the first "shadow" is $\xi_{\mathrm{P,0}}(v_i)$ and $\xi_{\mathrm{C,1}}(v_i)$, which depend on the velocity of the "parallel light". They determine the spatial boundaries of domain $D_n$ in the state space. The Minkowski's length of the "shadow" projected on $\Sigma_{\mathrm{ini}}$ is a Lorentz scalar, therefore we may calculate this value in the laboratory reference frame ($u=0$).
• Figure 5: Separation of domain $D_n$ and the physical meaning of the overlap function $\varphi_n(v_i)$.