Landauer Bound in the Context of Minimal Physical Principles: Meaning, Experimental Verification, Controversies and Perspectives.

The physical roots, interpretation, controversies, and precise meaning of the Landauer principle are surveyed. The Landauer principle is a physical principle defining the lower theoretical limit of energy consumption necessary for computation. It states that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat kBTln2 per a bit of information to its surroundings. The Landauer principle is discussed in the context of fundamental physical limiting principles, such as the Abbe diffraction limit, the Margolus-Levitin limit, and the Bekenstein limit. Synthesis of the Landauer bound with the Abbe, Margolus-Levitin, and Bekenstein limits yields the minimal time of computation, which scales as τmin~hkBT. Decreasing the temperature of a thermal bath will decrease the energy consumption of a single computation, but in parallel, it will slow the computation. The Landauer principle bridges John Archibald Wheeler's "it from bit" paradigm and thermodynamics. Experimental verifications of the Landauer principle are surveyed. The interrelation between thermodynamic and logical irreversibility is addressed. Generalization of the Landauer principle to quantum and non-equilibrium systems is addressed. The Landauer principle represents the powerful heuristic principle bridging physics, information theory, and computer engineering.

Fermat Principle, Ramsey Theory and Metamaterials.

Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the "inverse graph" is introduced and discussed. The total number of triangles in the "direct" (source) and "inverse" Ramsey optical graphs is the same. The applications of "Ramsey optics" are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested.

Shannon Entropy of Ramsey Graphs with up to Six Vertices.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and ß-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic ß-colored convex polygons with m ß-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sß are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph; Sß is, in turn, an average uncertainty to find the red ß-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Apple-like Shape of Freezing Paraffin Wax Droplets and Its Origin.

Paraffin wax stores energy in the form of latent heat at a nearly constant temperature during melting and releases this energy during solidification. This effect is used in industrial energy storage. At the same time, the possible deformation of even small volumes of material as a result of phase change is insufficiently studied. In this paper, the physical nature of such deformation, probably for the first time, is studied on the example of a droplet of paraffin wax. An unusual change in the shape of a melted droplet of paraffin wax placed on a relatively cold glass plate was observed in the laboratory experiments. As the droplet solidifies, its upper surface becomes nearly flat, and a dimple is formed in the center of this surface, making the droplet look like a fruit (pumpkins are more commonly shaped like this, but the authors prefer apples). A series of experiments, as well as physical and numerical modeling of the droplet's thermal state, taking into account the formation of a mushy zone between liquidus and solidus, made it possible to understand the role of gravity and gradual increase in viscosity and density of paraffin wax on changing the droplet shape and, in particular, to clarify the mechanism of formation of the dimple on its upper. It was shown that the mushy zone between the liquidus and solidus of the paraffin wax is responsible for the dimple formation.

Universality of Scaling Laws Governing Contact and Spreading Time Spans of Bouncing Liquid Marbles and its Physical Origin.

The impact of liquid marbles coated with a diversity of hydrophobic powders with various solid substrates, including hydrophobic, hydrophilic, and superhydrophobic ones, was investigated. The contact time of the bouncing marbles was studied. Universal scaling behavior of the contact time tc as a function of the Weber number (We) was established; the scaling law tc = tc(We) was independent of the kind of powder and the type of solid substrate. The total contact time consists of spreading time and retraction time. It is weakly dependent on We and this is true for all kinds of studied powders and substrates. This observation hints to the surface tension/inertia spring model governing the impact. By contrast, the spreading time ts scales as [Formula: see text], n = 0.28 - 0.30 ± 0.002. We relate the origin of this scaling law to the viscous dissipation occurring within the spreading marbles. The retraction time tr grows weakly with the Weber number. The scaling law was changed at threshold values of We ≅ 15-20. It is reasonable to explain this change with the breaking of the Leidenfrost regime of spreading under high values of We.

Ramsey theory and thermodynamics.

Re-shaping of thermodynamics with the graph theory and Ramsey theory is suggested. Maps built of thermodynamic states are addressed. Thermodynamic states may be attainable and non-attainable by the thermodynamic process in the system of constant mass. We address the following question how large should be a graph describing connections between discrete thermodynamic states to guarantee the appearance of thermodynamic cycles? The Ramsey theory supplies the answer to this question. Direct graphs emerging from the chains of irreversible thermodynamic processes are considered. In any complete directed graph, representing the thermodynamic states of the system the Hamiltonian path is found. Transitive thermodynamic tournaments are addressed. The entire transitive thermodynamic tournament built of irreversible processes does not contain a cycle of length 3, or in other words, the transitive thermodynamic tournament is acyclic and contains no directed thermodynamic cycles.

Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings.

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

Interfacial crystallization in the polyhedral liquid marbles.

HYPOTHESIS: We hypothesized that interfacial crystallization occurring within evaporated polyhedral liquid marbles may be controlled by hydrophilization of the polymer plates coating the marbles. The hypothesis was tested with polyhedral marbles coated with hydrophobized and cold plasma-hydrophilized PET (Polyethylene terephthalate) plates. EXPERIMENTS: Interfacial crystallization within polyhedral liquid marbles was investigated experimentally. Two types of polyhedral marbles filled with saturated saline were prepared: i) liquid marbles coated with hydrophobized PET plates (Marbles A); ii) liquid marbles coated with Janus PET plates, one facet of which was plasma hydrophilized and the other hydrophobized (Marbles B). The hydrophobized side of the PET plate was in contact with the saline solution, whereas, the hydrophilized facet contacted air. Crystallization occurring within the marbles under their evaporation was monitored in situ. FINDINGS: It was established that for both kinds of marbles, NaCl crystallization was initiated at the edges of the plates. NaCl crystallization on the hydrophobized PET surfaces was not registered. When Marbles B were evaporated, the outer hydrophilic side of the PET plates was coated by the saline creep process. For both kinds of marbles the process resulted in the formation of hollow shells built of PET plates and NaCl crystals. The thermodynamic explanation of the observed phenomena is suggested.

Hierarchical liquid marbles formed using floating hydrophobic powder and levitating water droplets.

HYPOTHESIS: Liquid marbles i.e. droplets coated by hydrophobic particles may be formed not only on the solid substrates but also on the floating layers of hydrophobic powders such as fluorinated fumed silica or polytetrafluoroethylene. EXPERIMENTS: Formation and growth of liquid marbles on fluorinated fumed silica or polytetrafluoroethylene powder floating on a heated water-vapor interface is reported. Marbles emerge from condensation of water droplets levitating above the powder layer. FINDINGS: The kinetics of the growth of droplets is reported. Growth of droplets results from three main mechanisms: water condensation, absorption of small droplets and merging of droplets with neighboring ones. Growing droplets are coated with the hydrophobic powder, eventually giving rise to the formation of stable liquid marbles. Formation of hierarchical liquid marbles is reported. Growth of liquid marbles emerging from water condensation follows the linear temporal dependence. A phenomenological model of the liquid marble growth is suggested.

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns.

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.

Branched droplet clusters and the Kramers theorem.

Scaling laws inherent for polymer molecules are checked for the linear and branched chains constituting two-dimensional (2D) levitating microdroplet clusters condensed above the locally heated layer of water. We demonstrate that the dimensionless averaged end-to-end distance of the droplet chain r[over ¯] normalized by the averaged distance between centers of the adjacent droplets l[over ¯] scales as r[over ¯]/l[over ¯]∼n^{0.76}, where n is the number of links in the chain, which is close to the power exponent ¾, predicted for 2D polymer chains with excluded volume in the dilution limit. The values of the dimensionless Kuhn length b[over ̃]≅2.12±0.015 and of the averaged absolute value of the bond angle of the droplet chains |θ|[over ¯]=22.0±0.5^{0} are determined. Using these values we demonstrate that the predictions of the Kramers theorem for the gyration radius of branched polymers are valid also for the branched droplets' chains. We discuss physical interactions that explain both the high value of the power exponent and the applicability of the Kramers theorem including the effects of the excluded volume, surrounding droplet monomers, and the prohibition of extreme values of the bond angle.

Effect of asymmetric cooling of sessile droplets on orientation of the freezing tip.

HYPOTHESIS: The shape of the "freezing tip" formed by the crystallization of water droplets demonstrated remarkable universality - no dependence on the cooling rate and physico-chemical properties of the substrate has been observed. At the same time, the spatial orientation of the freezing cone may be varied. We hypothesized that the orientation of the freezing tip is determined by the direction of heat flux at the base of the sessile droplet. This direction is expected to be changed when the substrate with a low thermal diffusivity is not cooled uniformly. EXPERIMENTS: We studied the freezing of water droplets placed on the inclined surface of wedges made from a variety of materials (polymers: Polymethylmethacrylate, Polytetrafluoroethylene, Polyurethane and metal: Titanium), which were cooled from below. The shape of the frozen droplets was controlled in situ. COMPUTATIONS: The computational model was suggested for the transient temperature field in the polymer wedge to determine a time variation of the local heat flux under the droplets. A comparison of numerical results and the measurements enabled us to confirm the aforementioned hypothesis relating the orientation of the freezing tips to the direction of the heat flux. FINDINGS: It was established that the orientation of the freezing cone axis depends on the location of the frozen droplet on the inclined surface of the wedge. Calculations of the transient temperature field of the wedge confirmed our hypothesis about the physical reason of the various spatial orientations of the freezing cones.

Rotating Minimal Thermodynamic Systems.

Minimal rotating thermodynamic systems are addressed. Particle m placed into the rotating symmetrical double-well potential (bowl), providing binary logical system is considered. The condition providing the transfer of the particle from one frictionless half-well to another, and, in this way, the possibility to record 1 bit of information is derived. The procedure of recording turns out to be irreversible; it is impossible to return the particle to its initial state under rotation about the same axis. The same rotating double-well system exerted to the thermal noise is considered. A minimal rotating thermal engine built of the rotating chamber, movable partition, and the particle confined within the chamber is treated. Rotation of the system displaces the partition, thus enabling erasing of one bit information. Erasing of 1 bit of information is due to the inertia (centrifugal force) acting on the partition. Isothermal expansion of the "minimal gas" expectedly gives rise to the Landauer bound. Compression of the "gas" with the rotation around the same axis is impossible and demands the additional axis of rotation. The interrelation between the possibility of recording/erasing information and the symmetry of the system is considered.

Magnetic Entropic Forces Emerging in the System of Elementary Magnets Exposed to the Magnetic Field.

A temperature dependent entropic force acting between the straight direct current I and the linear system (string with length of L) of N elementary non-interacting magnets/spins µâ†’ is reported. The system of elementary magnets is supposed to be in the thermal equilibrium with the infinite thermal bath T. The entropic force at large distance from the current scales as Fmagnen~1r3, where r is the distance between the edge of the string and the current I, and kB is the Boltzmann constant; (r≫L is adopted). The entropic magnetic force is the repulsion force. The entropic magnetic force scales as Fmagnen~1T, which is unusual for entropic forces. The effect of "entropic pressure" is predicted for the situation when the source of the magnetic field is embedded into the continuous media, comprising elementary magnets/spins. Interrelation between bulk and entropy magnetic forces is analyzed. Entropy forces acting on the 1D string of elementary magnets that exposed the magnetic field produced by the magnetic dipole are addressed.

Topology of eeg wave fronts.

Whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. This seemingly worthless sentence is an informal description of the hairy ball theorem, an invaluable mathematical weapon that has been proven useful to describe a variety of physical/biological processes/phenomena in terms of topology, rather than classical cause/effect relationships. In this paper we will focus on the electrical brain field-electroencephalogram (EEG). As a starting point we consider the recently-raised observation that, when electromagnetic oscillations propagate with a spherical wave front, there must be at least one point of the tangential components of the vector fields where the electromagnetic field vanishes. We show how this description holds also for the electric waves produced by the brain and detectable by EEG. Once located these zero-points in EEG traces, we confirm that they are able to modify the electric wave fronts detectable in the brain. This sheds new light on the functional features of a nonlinear, metastable nervous system at the edge of chaos, based on the neuroscientific model of Operational Architectonics of brain-mind functioning. As an example of practical application of this theorem, we provide testable previsions, suggesting the proper location of transcranial magnetic stimulation's coils to improve the clinical outcomes of drug-resistant epilepsy.

Interfacial crystallization at the intersection of thermodynamic and geometry.

Interfacial crystallization appears as a crucial stage in the numeral natural phenomena and technological applications, such as industry of semi-conductors and manufacturing of nano-whiskers. Interfacial aspects of heterogeneous crystallization are surveyed. The review is focused on the interplay of thermodynamic and geometric aspects of the interfacial crystallization. Thermodynamic considerations leading to the Wulff construction are discussed. Equilibrium shape of the crystallized particle in the contact with a foreign substrate giving rise to the Winterbottom construction is treated. The concept of equivalent equilibrium contact angle θeq is introduced. The equivalent contact angle θeq applicable for isotropic crystals does not depend neither on the volume of the crystallized particles nor on the external fields. Bulk contributions to the free energy of the particle such as the bulk heat release in the case of reactive contact or latent heat of crystallization do not influence the equivalent contact angle θeq. Application of the Winterbottom constructions for prediction of the shape of nanoparticles grown on solid substrates is treated. Thermodynamics of interfacial crystallization is discussed. The thermodynamic condition predicting when surface crystallization is thermodynamically favored over homogeneous (bulk) crystallization is supplied. This thermodynamic relation coincides with the condition prescribing the partial wetting of a solid by its melt. Interfacial crystallization in its relation to the "coffee-stain" effect, salt creeping and development of anti-icing surfaces is addressed. Interfacial aspects of epitaxial growth of crystals are considered. The current state-of-art in the field is reviewed.

Robust icephobic coating based on the spiky fluorinated Al2O3 particles.

Omniphobic and icephobic twin-scale surfaces based on the "urchin"-like fluorinated Al2O3 particles are presented. Combined effect of hierarchical topography and fluorination supplied to the surfaces omniphobic and icephobic properties. The study of the stability of the Cassie wetting state is reported. High apparent contact angles were accompanied with the low contact angle hysteresis and high stability of the Cassie air trapping wetting state. Time delay of the ice crystallization as high as [Formula: see text] min was established when compared to the ice formation on flat aluminum and non-fluorinated "urchin"-like surfaces. Crystallized water droplets formed on the reported nano-structured surfaces were easily blown out by the air jet with the velocity of [Formula: see text] m/s, (which is markedly lower than that common for exploitation of aircrafts and turbines). Heated "urchin"-like surfaces completely restored their omniphobic and icephobic surfaces after thawing. Qualitative analysis of water freezing is supplied.

Osmotic evolution of composite liquid marbles.

HYPOTHESIS: We hypothesized that the reported evolution (growth) of composite water marbles filled with saline water and coated with lycopodium dispersed in a thin layer of silicone oil is due to the osmotic mass transfer. The hypothesis is supported by the semi-empirical model of osmotic growth of small liquid marbles floating on distilled water. EXPERIMENTS: Saline composite, silicone oil-coated marbles floating on distilled water grew with time; whereas, composite marbles filled with distilled water floating on aqueous solutions of NaCl lost mass with time and shrunk. However, composite liquid marbles filled with saline water and floating on aqueous solutions of NaCl remained stable during 25 h of the laboratory experiment. FINDINGS: The reported findings are reasonably attributed to osmotic mass transport through the thin silicon layer filled with lycopodium particles coating the marbles, acting as an osmotic membrane. This is supported by the suggested model for the osmotic growth of marbles.

Survival of Virus Particles in Water Droplets: Hydrophobic Forces and Landauer's Principle.

Many small biological objects, such as viruses, survive in a water environment and cannot remain active in dry air without condensation of water vapor. From a physical point of view, these objects belong to the mesoscale, where small thermal fluctuations with the characteristic kinetic energy of kBT (where kB is the Boltzmann's constant and T is the absolute temperature) play a significant role. The self-assembly of viruses, including protein folding and the formation of a protein capsid and lipid bilayer membrane, is controlled by hydrophobic forces (i.e., the repulsing forces between hydrophobic particles and regions of molecules) in a water environment. Hydrophobic forces are entropic, and they are driven by a system's tendency to attain the maximum disordered state. On the other hand, in information systems, entropic forces are responsible for erasing information, if the energy barrier between two states of a switch is on the order of kBT, which is referred to as Landauer's principle. We treated hydrophobic interactions responsible for the self-assembly of viruses as an information-processing mechanism. We further showed a similarity of these submicron-scale processes with the self-assembly in colloidal crystals, droplet clusters, and liquid marbles.

Directional Droplet Transport Mediated by Circular Groove Arrays. Part II: Theory of Effect.

In the first part of this research, we reported the experimental study of the drop impact on the superhydrophobic circular groove arrays, which resulted in a directional droplet transport. In the second part, we further explored the influence of the Weber number (We), ridge height (H), and the deviation distance (r) between the impacting point and the center of curvature on the lateral offset distance (ΔL) of bouncing drops. The suggested theoretical analysis is in reasonable agreement with the experimental observations. We demonstrate that a Cassie-Wenzel wetting transition occurred within the microstructures of the relief under the threshold Weber number, for example, We ≅ 19-25, which switched the nature of drop bouncing. The dynamic pressure plays a decisive role in the directional droplet transport. The reported investigation may shed light on the solid-liquid interactions occurring on the patterned hierarchical surfaces and open up new opportunities for directional droplet transportation.