Work as a memory record

The possibility of a controlled manipulation with molecules at the nanoscale allows us to gain net work from thermal energy, although this seems to be in contradiction to the second law of thermodynamics. Any manipulation, however, causes some memory records somewhere in the system's surroundings. To complete the thermodynamic cycle, these records must be reset, which costs energy that cancels the previous gain. An important memory record may also be the final state of the work reservoir. This memory record is not reset. Nevertheless, it is rewritten and the associated memory erased whenever the state of the work reservoir is changed during the cycle repeating. The question is, what is the cost of this memory erasure. To answer it, we study a hypothetical cycle in which all memory records are reset except the state of the work reservoir alone, and the ensemble average of the work extracted from an equilibrium heat reservoir during this cycle, $\langle W\rangle$, is positive. It is shown that a strong information coupling of the system and the work reservoir then influences the system's dynamics so much that the cycle repeat may exhibit quite different behavior. Especially, it can run reversely and decrease energy in the work reservoir. It implies that even if the memory erasure is a natural part of the process, it costs energy in accord with information thermodynamics. At the nanoscale, this energy cost may manifest as a process obeying the fluctuation theorem.

Figure 1

A half-working Szilard engine that works without any measurement or feedback. It seems that—when repeating the cycle many times—it may accumulate positive work. The position of the weight is a memory that records which process has passed.

Figure 2

A toy mechanical illustration of an extracting barrier. A thermalized pendulum is the source of energy; a ball (at $t=0$ at rest) obtains an energy $E_0$, goes up, and transmits a part of its energy to a lift raising a weight if $E_0$ exceeds $\mathrm{\Delta }E$. The transfer of energy to the ball is random, so the final position of the lift is random too: either it is at a height corresponding to the energy $\mathrm{\Delta }E$ or is at the zero position (no shift).

Figure 3

After the first thermalization, the system is above the barrier (the subspace of the state space ${\mathrm{\Gamma }}_u$) and supplies the energy $\mathrm{\Delta }E$ to the weight. Suppose that after the second thermalization the system is at its lowest state. The construction of the whole engine should guarantee that the system rests in this state during the cycle and no exchange of energy with the weight happens. It is, however, impossible since the state $(x_0,w)$ is the final state of the state $(x_n,w_0)$ and cannot be simultaneously the final state of $(x_0,w)$. The energy of the system hence must increase and the weight shift down.

Figure 4

Presentation of the system dynamics as a mapping between the initial, ${\mathrm{\Gamma }}_0$, and final, $\mathrm{\Gamma }$, state space of X. The initial and final state spaces of the system A+WR are presented too: its final state (e.g., $\alpha ,\beta$) is the memory record. A “microscopic observer” knowing details of the final state space of X knows the final state $s$ but cannot decide which history is true—a or b. A “macroscopic observer” does not know the final state of X but knows the value of the memory record, say, $\beta$. This implies that “it” knows that the history is b and the final (initial) state of X belongs in ${\mathrm{\Gamma }}_{\beta }$ (${\mathrm{\Gamma }}_{0\beta }$).

Figure 5

Two stages of the process. In the first stage, the contact of the system X with a heat reservoir adjusts random initial conditions at $t=0$. Then the system evolves in a deterministic way till $t=\tau$. The system dynamic is interconnected with the auxiliary device A and the work reservoir WR. If this interconnection is organized in such a special way that A returns to its beginning configuration, then the process defines the adiabatic accessibility at microscopic level.