Landauer's Principle
Encyclopedia
Landauer's Principle, first argued in 1961 by Rolf Landauer
of IBM
, 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". (Bennett 2003).
The Landauer's Principle describes the Landauer limit, which is the minimum possible amount of energy required to change one bit of information, as follows:
At 25 °C (room temperature, or 298.15 kelvins), the Landauer limit represents an energy of approximately 0.0178 electron volt. Theoretically, roomtemperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second with only 2.85 trillionths of a watt of power being expended in the memory media.
If no information is erased, computation may in principle be achieved which is thermodynamically reversible
, and require no release of heat. This has led to considerable interest in the study of reversible computing
.
The principle is widely accepted as physical law
; but in recent years it has been challenged, notably in Earman and Norton (1998), and subsequently in Shenker (2000) and Norton (2004), and then defended by Bennett (2003) and Ladyman et al. (2007).
—which states that the entropy of a closed 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 originally (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.) So, to avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment at some given temperature T, requiring that energy E = ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit
of logical information is lost, the amount of entropy generated is at least k ln 2, and so the energy that must eventually be emitted to the environment is E ≥ kT ln 2.
In 2003 Weiss and Weiss came on a background of psychometric data and theoretical considerations to the conclusion that information processing by the brain has to be based on Landauer's principle. In 2008 this has been empirically confirmed by a group of neurobiologists.
This expression for the minimum energy dissipation from a logically irreversible binary operation was first suggested by John von Neumann
, but it was first rigorously justified (and with important limits to its applicability stated) by Landauer. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.
Rolf Landauer
Rolf William Landauer was an IBM physicist who in 1961 argued that when information is lost in an irreversible circuit, the information becomes entropy and an associated amount of energy is dissipated as heat...
of IBM
IBM
International Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
, holds that "any logically irreversible manipulation of information, such as the erasure of a bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
or the merging of two computation paths, must be accompanied by a corresponding entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
increase in non-information bearing degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
of the information processing apparatus or its environment". (Bennett 2003).
The Landauer's Principle describes the Landauer limit, which is the minimum possible amount of energy required to change one bit of information, as follows:
- Where…
- is the Boltzmann constant
- is the absolute temperature of the circuit in kelvinKelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
s - is the natural logarithmNatural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
of 2, which is approximately 0.69315
At 25 °C (room temperature, or 298.15 kelvins), the Landauer limit represents an energy of approximately 0.0178 electron volt. Theoretically, roomtemperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second with only 2.85 trillionths of a watt of power being expended in the memory media.
If no information is erased, computation may in principle be achieved which is thermodynamically reversible
Reversible process (thermodynamics)
In thermodynamics, a reversible process, or reversible cycle if the process is cyclic, is a process that can be "reversed" by means of infinitesimal changes in some property of the system without loss or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic...
, and require no release of heat. This has led to considerable interest in the study of reversible computing
Reversible computing
Reversible computing is a model of computing where the computational process to some extent is reversible, i.e., time-invertible. A necessary condition for reversibility of a computational model is that the transition function mapping states to their successors at a given later time should be...
.
The principle is widely accepted as physical law
Physical law
A physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...
; but in recent years it has been challenged, notably in Earman and Norton (1998), and subsequently in Shenker (2000) and Norton (2004), and then defended by Bennett (2003) and Ladyman et al. (2007).
Rationale
Landauer's principle can be understood to be a simple logical consequence of the second law of thermodynamicsSecond law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
—which states that the entropy of a closed system
Closed system
-In physics:In thermodynamics, a closed system can exchange energy , but not matter, with its surroundings.In contrast, an isolated system cannot exchange any of heat, work, or matter with the surroundings, while an open system can exchange all of heat, work and matter.For a simple system, with...
cannot decrease—together with the definition of thermodynamic temperature
Thermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
. 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 originally (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
Holographic principle
The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
is correct, then physical systems with finite surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...
have a finite maximum entropy; but regardless of the truth of the holographic principle, quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
dictates that the entropy of systems with finite radius and energy is finite.) So, to avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment at some given temperature T, requiring that energy E = ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
of logical information is lost, the amount of entropy generated is at least k ln 2, and so the energy that must eventually be emitted to the environment is E ≥ kT ln 2.
In 2003 Weiss and Weiss came on a background of psychometric data and theoretical considerations to the conclusion that information processing by the brain has to be based on Landauer's principle. In 2008 this has been empirically confirmed by a group of neurobiologists.
This expression for the minimum energy dissipation from a logically irreversible binary operation was first suggested by John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, but it was first rigorously justified (and with important limits to its applicability stated) by Landauer. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.
External links
- Introductory article on Landauer's principle and reversible computing
- Maroney, O.J.E. " Information Processing and Thermodynamic Entropy" The Stanford Encyclopedia of Philosophy.
- Eurekalert.org: “Magnetic memory and logic could achieve ultimate energy efficiency”, 1-Jul-2011