1. Introduction
Landauer principle is one of the limiting physical principles, which constraint behavior of physical systems. There exist fundamental laws and principles setting the limits of physical systems. These laws do not predict or describe behavior of physical/engineering systems, but limit, restrict their functioning. A realistic natural/engineering system can only provide limited functionalities because its performance is physically constrained by some basic principles [
1]. Some of these limits are engineering ones. For example, a key engineering bottleneck for the development of new generations of computers today is integrated circuits manufacture, which packs billions of transistors and wires in several
of silicon with astronomically low defect rates [
2]. Another engineering constraint is imposed by limits on individual interconnects [
2]. Despite the doubling of transistor density with Moore’s law, semiconductor integrated circuits (ICs) would not work without fast and dense interconnects. Metallic wires can be either fast or dense, but not both at the same time - smaller cross-section increases electrical resistance, while greater height or width increase parasitic capacitance with neighboring wires (wire delay grows with RC) [
2]. Other constraints limiting the operation of physical (natural or engineering) systems are fundamental ones, and they emerge from the deepest foundations of physics. Limiting physical principles appeared in physics relatively late. It seems, that the first limiting principle historically was the Abbe diffraction limit, discovered in 1873, which states that light with wavelength
, traveling in a medium with refractive index
n and converging to a spot with half-angle
will have a minimum resolvable distance of
d, supplied by Eq. 1:
The Abbe diffraction limit is the maximum resolution possible for a theoretically perfect, or ideal, optical system [
3,
4]. Thus, it is not the engineering, but the fundamental physical principle. The Abbe diffraction limit arises from the idea that the image arises from a double diffraction process [
3,
4]. In spite of the fact that the Abbe diffraction limit is rooted in the classical physics, the role of the limiting principles in the realm of the classical physics is more than modest. The situation changed dramatically within the modern physics. In the relativity the speed of light in vacuum, labeled
c, is a universal physical constant that is ca. 300,000 kilometers per second, and according to the special theory of relativity,
c is the upper limit for the speed at which conventional matter or energy (and, consequently, any signal carrying information) can travel through space [
6,
7]. It is impossible for signals or energy to travel faster than
c. The speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of the observer; thus, enabling Einstein synchronization procedure for the clocks [
6,
7]. The limiting status of the speed of light in vacuum was intensively disputed in last decades and theories assuming a varying speed of light have been proposed as an alternative way of solving several standard cosmological problems [
8,
9]. Recent observational hints that the fine structure constant may have varied over cosmological scales have given impetus to these theories [
8,
9]. Theories in which a light of speed of vacuum appeared as an emerging physical value were suggested [
9]. We adopt unequivocally the limiting status of a speed of light in vacuum
c and demonstrate that this status generates other limiting physical principles, and just this status gives rise to consequences emerging from the Landauer principle.
The main limiting principle of the quantum mechanics is the Heisenberg uncertainty principle. It states that there is a limit to the precision with which certain pairs of physical properties, such as position
x and momentum
p (or time
t and energy
E), can be simultaneously measured. In other words, the more accurately speaking one property is measured, the less accurately the other property can be established (see Eq. 2 and Eq. 3):
where
and
are standard deviations of the position, momentum, time and energy correspondingly and
is the reduced Planck constant [
10,
11]. The time-energy uncertainty principle, supplied with Eq. 3 needs more detailed discussion to be supplied below in the context of Mandelstam-Tamm and Margolus-Levitin bounding principles.
The limiting value of the light propagating in vacuum
c combined with the Heisenberg uncertainty principle yield together the Bremermann’s limit, which supplies a limit on the maximum rate of computation that can be achieved in a self-contained system [
12]. Bremermann’s limit is derived from Einstein’s mass-energy equivalency and the Heisenberg uncertainty principle, and is
bits per second per kilogram of the computational system [
12]. Consider that the Bremermann limit is built of the fundamental physical constants only.
Quantum mechanics gives rise also to the Mandelstam-Tamm and Margolus-Levitin limiting principles [
13,
14]. The Mandelstam-Tamm quantum speed limit states that the time it takes for an isolated quantum system to evolve between two fully distinguishable states is given by Eq. 4:
where
is the energy uncertainty. The Margolus-Levitin limiting principle supplies a surprising result, predicting the maximum speed of dynamical evolution of the system [
15]. The Margolus-Levitin limiting principle supplies the minimal time it takes for the physical system to evolve into orthogonal state (labeled
. It should be emphasized that this minimal time
depends only on the system average energy minus its ground state (denoted
, and not on the energy uncertainty
as it follows from Eq. 4 [
15]. To simplify the formulae we choose zero of energy in such a way that
so that the Margolus-Levitin limiting principle yields for the minimal time bound denoted
Eq. 5:
Another important fundamental limiting principle is supplied by the Bekenstein bound [
16]. Bekenstein bound defines an upper limit on the entropy
S, which can be confined within a given finite region of space which has a finite amount of energy—or conversely, the maximal amount of information required to perfectly describe a given physical system down to the quantum level [
16]. The bound value of entropy
S is given by Eq. 6:
where
R is the radius of a sphere that can enclose the given system,
E is the total mass–energy including any rest masses [
16]. We will discuss below the Margolus-Levitin and the Bekenstein bounds in their relation to the Landauer principle.
3. Conclusions
The physical roots, justification, interpretation, controversies and precise meaning of the Landauer principle remain obscure, in spite of the fact that they were exposed to the turbulent and spirited discussion in the last decades. Landauer’s principle (or Landauer bound), suggested by Rolf Landauer in 1961, is a physical principle pertaining to the lower theoretical limit of energy consumption of computation [
26,
27,
28,
29]. It states that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat
per a bit information to its surrounding. The Landauer Principle is discussed in the context of other fundamental physical limiting principles, such as the Abbe diffraction limit, the Margolus-Levitin limit and the Bekenstein limit [
15,
16]. We demonstrate the synthesis of the Landauer bound with the Abbe, Margolus-Levitin limit and the Bekenstein limit quite surprisingly yields the minimal time of computation, which scales as
(where
h and
are the Planck and Boltzmann constants correspondingly), which is exactly the Planck-Boltzmann thermalization time [
36,
41]. This result leads to a very important conclusion: decrease in a temperature of a thermal bath will decrease the energy consumption of a single computation, but in parallel, it will slow the computation. The relation between the Landauer bound and the Szilard minimal engine is discussed.
The Landauer principle bridges between John Archibald Wheeler “it from bit” paradigm and thermodynamics [
63,
73,
74]. This bridge yields the mass energy-information principle, enables calculation of the informational capacity of the Universe and provides a fresh glance on the dark matter problem [
64,
65,
66,
67,
68,
69]. The Landauer Principle may serve as a basis for unification of physical theories, enabling the united approach to the informational content of fields and particles. Generalization of the Landauer principle to the quantum and non-equilibrium systems is addressed [
44,
45]. The relativistic aspects of the Landauer principle are discussed. Engineering applications of the Landauer principle in the development of optimal computational protocols are considered [
37,
43,118]. Experimental verifications of the Landauer Principle are surveyed [
46,
59]. Interrelation between thermodynamic and logical irreversibility is addressed. Non-trivial relationship between the Landauer Principle and the Second Law of thermodynamic is considered [
113]. Objections and criticism of the Landauer Principle are discussed [
101,
102,
107]. We conclude that the Landauer Principle represents the powerful heuristic principle bridging the fundamental physics, information theory and computer engineering. It is suggested. that the Landauer Principle may serve as a cornerstone of the axiomatic thermodynamic.