Landauer's principle
Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that "any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment". Another way of phrasing Landauer's principle is that if an observer loses information about a physical system, heat is generated and the observer loses the ability to extract useful work from that  system. A so-called logically reversible computation, in which no information is erased, may in principle be carried out without releasing any heat. This has led to considerable interest in the study of reversible computing. Indeed, without reversible computing, increases in the number of computations per joule of energy dissipated must eventually come to a halt. If Koomey's law continues to hold, the limit implied by Landauer's principle would be reached around the year 2050. At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0175 eV, or 2.805 zJ. Theoretically , room-temperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second (1 Gbit/s) with energy being converted to heat in the memory media at the rate of only 2.805 trillionths of a watt (that is, at a rate of only 2.805 pJ/s). Modern computers use millions of times as much energy per second. History. Rolf Landauer first proposed the principle in 1961 while working at IBM. He justified and stated important limits to an earlier conjecture by John von Neumann. For this reason, it is sometimes referred to  as being simply the Landauer bound or Landauer limit. In 2011, the principle was generalized to show that while information erasure requires an increase in entropy, this increase could theoretically occur at no energy cost. Instead, the cost can be taken in another conserved quantity, such as angular momentum. In a 2012 article published in "Nature", a team of physicists from the École normale supérieure de Lyon, University of Augsburg and the University of Kaiserslautern described that for the first time they have measured the tiny amount of heat released when an individual bit of data is erased. In 2014 , physical experiments tested Landauer's principle and confirmed its predictions. In 2016, researchers used a laser probe to measure the amount of energy dissipation that resulted when a nanomagnetic bit flipped from off to on. Flipping the bit required 26 millielectronvolts (4.2 zeptojoules). A 2018 article published in "Nature Physics" features a Landauer erasure performed at cryogenic temperatures on an array of high-spin ("S" = 10) quantum molecular magnets. The array is made to act as a spin register where each nanomagnet encodes a single bit of information. The experiment has laid the foundations for the extension of the  validity of the Landauer principle to the quantum realm. Owing to the fast dynamics and low "inertia" of the single spins used in the experiment, the researchers also showed how an erasure operation can be carried out at the lowest possible thermodynamic cost—that imposed by the Landauer principle—and at a high speed. Rationale. Landauer's principle can be understood to be a simple logical consequence of the second law of thermodynamics—which states that the entropy of an isolated system cannot decrease—together with the definition of thermodynamic temperature. For, if the number of possible logical states  of a computation were to decrease as the computation proceeded forward (logical irreversibility), this would constitute a forbidden decrease of entropy, unless the number of possible physical states corresponding to each logical state were to simultaneously increase by at least a compensating amount, so that the total number of possible physical states was no smaller than it was originally (i.e. total entropy has not decreased). Yet, an increase in the number of physical states corresponding to each logical state means that, for an observer who is keeping track of the logical state of the system but not the physical state  (for example an "observer" consisting of the computer itself), the number of possible physical states has increased; in other words, entropy has increased from the point of view of this observer. The maximum entropy of a bounded physical system is finite. (If the holographic principle is correct, then physical systems with finite surface area have a finite maximum entropy; but regardless of the truth of the holographic principle, quantum field theory dictates that the entropy of systems with finite radius and energy is finite due to the Bekenstein bound.) To avoid reaching this maximum over the course of an extended  computation, entropy must eventually be expelled to an outside environment. Equation. Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the "Landauer limit": where formula_2 is the Boltzmann constant (approximately 1.38×10 J/K), formula_3 is the temperature of the heat sink in kelvins, and formula_4 is the natural logarithm of 2 (approximately 0.69315). After setting formula_3 equal to room temperature 20 °C (293.15 K), we can get the Landauer limit of 0.0175 eV (2.805 zJ) per bit erased. The equation can be deduced from Boltzmann's  entropy formula (formula_6), considering that formula_7 is the number of states of the system, which in the case of a bit is 2, and the entropy formula_8 is defined as formula_9. So the operation of erasing a single bit increases the entropy of a value of at least formula_10, emitting in the environment a quantity of energy equal or greater than formula_11. Challenges. The principle is widely accepted as physical law, but in recent years it has been challenged for using circular reasoning and faulty assumptions, notably in Earman and Norton (1998), and subsequently in Shenker (2000) and Norton (2004 , 2011), and defended by Bennett (2003), Ladyman et al. (2007), and by Jordan and Manikandan (2019). On the other hand, recent advances in non-equilibrium statistical physics have established that there is no a priori relationship between logical and thermodynamic reversibility. It is possible that a physical process is logically reversible but thermodynamically irreversible. It is also possible that a physical process is logically irreversible but thermodynamically reversible. At best, the benefits of implementing a computation with a logically reversible system are nuanced. In 2016, researchers at the University of Perugia claimed to have demonstrated a violation of Landauer’s  principle. However, according to Laszlo Kish (2016), their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".