# Fundamental Limits in Dissipative Processes during Computation

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## Abstract

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## 1. Introduction

## 2. Binary Switches

#### 2.1. Energy Dissipation in Charge-Based Switch Devices

#### 2.2. The Physics of Binary Switches

## 3. Fundamental Energy Limits in Binary Switches

#### 3.1. Combinational Switches and Logic Gates

#### 3.2. Sequential Switches and Memory Devices

#### 3.2.1. The Reset Operation

#### 3.2.2. The Switch Operation

#### 3.2.3. Memory Preservation

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Universal logic gate NAND. (

**Left**) The logic output OUT is here associated with an electric voltage value across a simple circuit: high-V corresponds to the logic state 1 and low-V corresponds to the logic state 0. Binary switches are represented here as mechanical switches that can assume the logic state “0” (physical state open) or the logic state “1” (physical state close). When both the switches are in the close position, the circuit behaves as a simple conductor and the voltage ${V}_{OUT}$ position assumes the value low-V. (

**Right**) Logic gate NAND implemented using transistors as binary switches. In this case, both the input logic state and the output logic state are physically encoded into electrical voltages.

**Figure 2.**Potential functions $U\left(x\right)$ with the associated probability densities $p(x,t)$. (

**Left column**) potential function $U\left(x\right)$ for the combinational switch. Here, the threshold ${x}_{TH}=1.2$. Upper graph: $p(x,t)$ associated with the logic state 0. Here, ${p}_{0}\simeq 1$ and ${p}_{1}\simeq 0$. Lower graph: $p(x,t)$ associated with the logic state 1. Here, ${p}_{0}\simeq 0$ and ${p}_{1}\simeq 1$. (

**Right column**) potential function $U\left(x\right)$ for the sequential switch. Here, the threshold ${x}_{TH}=0$. Upper graph: $p(x,t)$ associated with the logic state 0. Here, ${p}_{0}\simeq 1$ and ${p}_{1}\simeq 0$. Lower graph: $p(x,t)$ associated with the logic state 1. Here, ${p}_{0}\simeq 0$ and ${p}_{1}\simeq 1$.

**Figure 3.**Average heat production during the cycled combinational switch operation as a function of protocol time ${\tau}_{p}$. By increasing the protocol time, the produced heat decreases following a power law. The average heat Q is shown here in ${k}_{B}T$ units, where ${k}_{B}$ is the Boltzmann constant and T is the room temperature.

**Figure 4.**Reset to 0 operation. (

**Left**) probability density $p(x,t)$ at equilibrium. In this condition, ${p}_{0}={p}_{1}=0.5$. (

**Right**) probability density $p(x,t)$ after the reset operation. In this condition, ${p}_{0}\simeq 1$ and ${p}_{1}\simeq 0$. During the reset operation, the system entropy decreases by an amount $\Delta S=-{k}_{B}log2$. The same amount of entropy change is required for the reset to 1 operation.

**Figure 5.**Panels from left to right show the memory-loss mechanism when the bit 1 is initially stored. The curves give a qualitative time evolution of $p(x,t)$ as the relaxation to equilibrium process takes place.

**Figure 6.**Experimental results of produced heat for a single refresh operation as a function of the protocol time ${t}_{p}$. By increasing ${t}_{p}$, the produced heat tends to the lower bound $Q=-T\Delta S$ represented here by the dashed line. The solid squares represent the heat from the experiment, while the solid line is the fit with the Zener dissipative model.

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**MDPI and ACS Style**

Chiucchiú, D.; Diamantini, M.C.; López-Suárez, M.; Neri, I.; Gammaitoni, L. Fundamental Limits in Dissipative Processes during Computation. *Entropy* **2019**, *21*, 822.
https://doi.org/10.3390/e21090822

**AMA Style**

Chiucchiú D, Diamantini MC, López-Suárez M, Neri I, Gammaitoni L. Fundamental Limits in Dissipative Processes during Computation. *Entropy*. 2019; 21(9):822.
https://doi.org/10.3390/e21090822

**Chicago/Turabian Style**

Chiucchiú, Davide, Maria Cristina Diamantini, Miquel López-Suárez, Igor Neri, and Luca Gammaitoni. 2019. "Fundamental Limits in Dissipative Processes during Computation" *Entropy* 21, no. 9: 822.
https://doi.org/10.3390/e21090822