Energy {heat, energy}| {thermal energy} can be total random translational kinetic energy and total potential energy, in all directions, that holds atoms apart. Heat as kinetic energy raises temperature. Heat as potential energy makes solid into liquid and liquid into gas.
work
Heat energy has no direction. Heat energy in solids or liquids can do no work.
kinetic energy
Heat translational kinetic energy equals number of molecules times temperature.
potential energy
Solids have little heat potential energy. Liquids have some heat potential energy. Gases have much heat potential energy.
Drying, oxidizing, or reducing can use temperature just below melting {calcine}|.
Heat tends to push molecules apart {expansion, matter}| {thermal expansion} {heat expansion}. Temperature increase adds random translational kinetic energy to material and makes molecules collide faster, so molecules spread more. Material molecules have attractive forces, which resist random motion.
coefficient
Higher temperature makes material volume bigger by a ratio {coefficient of volume expansion} {volume-expansion coefficient}. Higher temperature makes material length longer by a ratio {coefficient of linear expansion} {linear-expansion coefficient}. Length change dL equals length L times temperature change dT times linear-expansion coefficient c: dL = c * L * dT. Volume change dV equals volume V times temperature change dT times volume-expansion coefficient c: dV = c * V * dT.
coefficient: gas
All gases have same volume-expansion coefficient, because gases approximate ideal gas.
examples
Road cracks, erosion, and thermostats with bimetallic strips illustrate thermal expansion.
Higher pressure forces most-substance molecules together and tends to make molecules go to lower potential energy {Joule-Thompson effect}. Expanding gas cools gas, as random translational kinetic energy changes into random potential energy.
state
If material is under pressure, state change happens at higher temperature. Higher temperature makes more kinetic energy to overcome higher pressure that keeps molecules together.
ice
Ice is less dense than water, so ice tends to melt under higher pressure. For example, pressure of ice-skate blade melts ice under blade to allow skating. If pressure on melted ice decreases, ice freezes again {regelation, pressure}.
altitude
Making cake at high altitude requires higher temperature, because lower air pressure makes air hold less heat.
Molecule motions and collisions make average random translational kinetic energy {temperature, heat}|. Average gas-molecule velocity at room temperature is 500 meters per second.
Materials can have no kinetic energy and no heat potential energy {absolute zero temperature}|, at -273.16 degrees Celsius.
Temperature scales {Celsius, temperature scale}| {centigrade} can set water freezing point to 0 C and boiling point to 100 C, at sea level.
Temperature scales {Kelvin, temperature}| can set water freezing point to 273 K and boiling point to 373 K, at sea level.
Surface temperature is proportional to energy emitted per unit area {Stefan's law} {Stefan law}.
Chemical reactions, engines, and collisions have force, energy, and heat transfers {thermodynamics}|.
heat
Energy transfers use work, through directed kinetic energy, or heat, through temperature change or state change. Friction changes some directed energy into random energy and increases temperature. Systems can minimize friction by slowing and by using lubricants.
comparison
Thermodynamics is about extensive quantities. Statistical mechanics is about intensive quantities. Thermodynamic quantities are number of moles times Avogadro's number times corresponding statistical-mechanics quantity. Molecular-property time averages give observable thermodynamic properties.
potentials
The six thermodynamic potentials are baryon-number density, total mass-energy density, isotropic pressure, temperature, entropy per baryon, and baryon chemical potential. Rest frame is stationary or moving fluid. Baryon number density and entropy per baryon determine composition. Baryon number is constant in fluid, because density is constant, so gradient equals zero. Systems can only create entropy, not destroy it. Shock waves increase entropy. Heat flows increase or decrease entropy.
Material transport {heat transport} properties, such as electric conductivity, thermal conductivity, viscosity, diffusion, effusion, and dissolution, depend on molecular properties such as temperature, pressure, collision frequency, and kinetic-energy range.
Heat flows have laws {thermodynamics laws}. When heat becomes another energy type or another energy type becomes heat, total energy does not change {energy conservation, first law} {first law of thermodynamics}. Heat flows from objects with higher temperature to objects with lower temperature, and energy must make heat flow from cold object to hot object {second law of thermodynamics}. Entropy is zero at absolute zero temperature {third law of thermodynamics}, because random motion is zero and system has complete order. Two systems in thermal equilibrium with third system have same temperature {zeroth law of thermodynamics}.
Systems react to change, such as energy change, to oppose further change {Le Chatelier's principle} {Le Chatelier principle}. As system resists change, directed work energy becomes random translational kinetic energy, through temperature and pressure change.
Systems with energy flows can have steady or periodic flow {steady state, thermodynamics}, rather than reach equilibrium. Movement rate or flux depends on gradient or force, so flow rate equals force or gradient sum. Steady states are irreversible thermodynamically. Entropy minimizes, because systems with forces or gradients can reduce entropy.
Perhaps, motion never slows {perpetual motion}|. Perpetual motion of first kind violates extended Le Chatelier's principle. Perpetual motion of second kind violates extended Le Chatelier's principle. Perpetual motion of third kind violates the principle that there must always be friction.
In heat exchange, heat lost by object equals heat gained by other object {conservation of heat energy} {heat energy conservation}| {law of heat exchange} {heat exchange law}.
Energy exchange can change potential energy, translational kinetic energy, and heat energy and change pressure and volume {enthalpy}|. Enthalpy equals total system energy E plus product of pressure P times volume V: H = E + P*V. Pressure times volume is work. Under constant pressure or volume, enthalpy is heat that system makes. For solids or liquids, enthalpy equals energy, because volume does not change.
Systems have energy {free energy}| available to do work. Free energy is energy from order loss plus potential energy converted to kinetic energy.
purpose
Free energy can show if process is spontaneous.
heat energy
Temperature times entropy is heat energy taken from surroundings.
work
Pressure times volume is work on system.
Helmholtz free energy
For constant temperature, free energy {Helmholtz free energy} is system energy minus heat energy: E - S*T.
Gibbs free energy
For constant pressure and temperature and changed volume, free energy {Gibbs free energy} is Helmholtz free energy plus work energy: E - S*T + P*V. Gibbs free energy G is enthalpy H minus temperature T times entropy S: G = H - T*S. Gibbs free energy is net work that system can do.
Arrhenius free energy
For changed temperature, free energy {Arrhenius free energy} is net work that system can do.
chemical potential
Gibbs free energy per mole u, the chemical potential, changes with absolute temperature T and mole fraction x: u = u0 + R * T * ln(x), where R is gas constant. Gibbs free energy per mole u changes with absolute temperature T and partial pressure P: u = u0 + R * T * ln(P).
free energy change
If system is not in equilibrium, something flows from higher to lower chemical potential. Free-energy change is negative. System changes spontaneously. However, spontaneous change does not happen if no pathway exists for energy change. To minimize free energy, system can lower potential energy, by reducing pressure, or increase entropy, by increasing temperature.
Isolated systems can have no work from outside. No energy transfers in or out of closed systems. Only entropy changes affect free energy.
Isothermal systems have only work and have no entropy change, because temperature is constant.
If temperature is low, entropy is small, so reaction makes heat to lower potential energy. If temperature is high, entropy is more important, and reaction heat can be small or large. At low pressure, more gas can evolve.
free energy change: equilibrium constant
In chemical reactions, free-energy change depends on equilibrium constant. Free-energy change equals gas constant times absolute temperature times natural logarithm of equilibrium constant.
free energy change: substances
For reactants, substance chemical potential times substance moles subtracts from reactant free energy. For products, substance chemical potential times substance moles adds to product free energy. Free-energy change in systems with one substance equals chemical potential a times change in number n of moles: a * n.
Chemical-reaction product and reactant concentrations depend on free-energy changes. Free-energy change equals -R * T * ln((ap1^np1 * ap2^np2 * ... ) / (ar1^nr1 * ar2^nr2 * ... )), where R is gas constant, T is temperature, api is product chemical potential, npi is chemical-equation product number of moles, ari is reactant chemical potential, and nri is chemical-equation reactant number of moles. Chemical reactions, and all physical changes, are spontaneous if they release free energy.
reaction
To reverse reactions, second reaction, with more free energy change, must couple to reaction. Total free energy change then favors reverse reaction. Diffusion, evaporation, and solvation take energy from surroundings, or use their thermal energy, to drive other reactions.
Heat can exert force in direction and so do work {work, heat}|. Possible work energy is difference in heat energy between hotter region and colder region, which is available heat energy. Machines have ratio {efficiency, work} between work actually done and heat available or input work. Efficiency is high temperature Th minus low temperature Tc divided by high temperature: (Th - Tc) / Th. Engines have efficiency of 30%.
Ideal engines have four stages {Carnot cycle}|: isothermal heat gain, adiabatic gas expansion, isothermal heat loss, and adiabatic gas contraction.
Temperature increase causes material to increase random translational kinetic energy and so absorb heat {heat capacity}|. Material can absorb heat and gain random translational kinetic energy, so temperature rises. Heat capacity is heat needed to raise one gram of material one degree Celsius. Heat H equals mass m times heat capacity c times temperature change T: H = m * c * T.
factors
Heat capacity depends on material type. Chemicals can hold more or less heat depending on possible electric dipole states. Metal atoms have no vibrations and rotations. Metals have low heat capacity, because all heat goes into random translational motion, rather than into vibrations or rotations. Diatomic molecules are linear molecules. Diatomic molecules have medium heat capacity, because they have few vibrations and rotations. Water is triatomic, is asymmetric, and has hydrogen bonds between molecules. Water has high heat capacity. Large complex molecules in gasoline, clays, and ceramics have high heat capacity. Crystal structure can have chemical bonds, hydrogen bonds, van der Waals forces, or ionic bonds, allowing many vibration modes and high heat capacity.
material heat capacity divided by water heat capacity {specific heat}|.
In reactions {explosion}|, temperature increase can increase reaction rate, which then increases temperature, which then increases rate, and so on. Gas production increases rapidly. Gas propels outward from center if reaction makes heat more rapidly than heat can dissipate by thermal radiation or gas loss. Randomly moving molecules tend to bounce outward, because surface area is greater toward perimeter and smaller toward center. Explosions require heat to stay high enough to burn gas before gas can move far.
burning
Burning does not explode, because gas has unconfined gas expansion or has much thermal radiation, so heat spreads out by thermal radiation or gas loss faster than reaction makes heat.
Ignition can spread flame through flammable gas at subsonic speeds {deflagration}|, as heat diffuses through gas. Gas expands evenly.
Ignition can spread flame through flammable gas at supersonic speeds {detonation}|, because shock waves compress gas. Detonation causes engine knocking, because gas expands unevenly.
In explosion-like reactions {implosion reaction}|, gas amount can decrease as temperature increase increases reaction rate, which then increases temperature, which then increases rate, and so on, because gas is reactant and products are not gases.
Heat change can happen over time {heat flow}. Heat flow is from high-temperature region to low-temperature region. Heat flow converted to translational kinetic energy exerted in direction can do work. Engines use adiabatic and isothermal heat-flow stages to perform work.
Heat flow can have constant heat {adiabatic}|. Temperature goes up in one location and down in another location.
Heat flow can have constant pressure {isobaric}|, typically in systems open to atmosphere.
Heat flow can have constant temperature {isothermal}|. Heat flows into or out of heat sinks or sources.
In heat flow {conduction, heat}|, collisions among molecules can transfer translational random kinetic energy.
materials
Fluids are good heat conductors, because molecules move freely. Metals are good heat conductors, because electrons move freely. Diamonds have high thermal conductivity, because crystal vibrations transfer heat.
area
Conductive-heat flow rate increases as contact area increases. Adding fins to surfaces or roughing up surfaces increases surface area and conducts heat better.
temperature
Conductive-heat flow rate increases as temperature difference increases. At room temperature, good conductor, such as metal, feels cool to touch, because heat moves quickly away from warmer human body. Poor conductor, such as plastic or wool, feels neither cool nor warm. Steering wheel covers and seat covers reduce heat conductivity.
In heat flow {convection, heat}|, mass can move in another mass, as hotter fluid at lower density rises and cooler fluid at higher density falls or as masses mix, flow, or blow. Convection is non-random motion that transfers heat by mass movement. Convective-heat flow rate increases as temperature difference increases. Blowing on something to cool it uses convection. Radiators use convection.
Electromagnetic-radiation emission or absorption {radiation, heat}| can transfer heat.
fire
Fire is electromagnetic radiation, emitting infrared and visible light from excited atoms in hot gas. Other objects can absorb radiation energy from fire and become hotter.
infrared
Heat radiation is typically infrared radiation. Infrared radiation is high in materials above 100 degrees Celsius.
color
White or shiny surfaces do not absorb radiation well, reflect radiation back into themselves at surfaces, and do not radiate at all frequencies well. Black surfaces absorb radiation well, do not reflect radiation back into themselves at surfaces, and radiate at all frequencies well.
Ideal objects {black body} can emit maximum heat radiation and have Planck distribution of radiation wavelengths and energies {black-body radiation}|.
Gas molecules move randomly, have elastic collisions, are point-like, and have no interactions {kinetic theory}|. Ideal gases follow kinetic theory. Gas molecules have cross-sectional area, and hydrogen bonds and van der Waals forces make molecules slightly attract, so real gas molecules do not move completely randomly and have somewhat inelastic collisions.
molecular collisions
In gases, one cubic centimeter has 10^28 molecular collisions per second. Collision frequency increases as mass decreases, temperature increases, cross-sectional area increases, and density increases.
molecular velocity
Gas-molecule collisions distribute speeds and directions. Molecular-velocity distributions are Boltzmann distributions. Some molecules have low velocity. Most molecules are near average velocity. Few molecules have very high velocities. Average gas-molecule velocity at room temperature is 500 meters per second. Molecular velocity increases as mass decreases or temperature increases.
Maxwell envisioned a demon {Maxwell's demon} {Maxwell demon} that can see particle motions and act on particles individually, so perpetual motion of second kind can happen. However, demon, light, and energy are all system parts, so perpetual motion cannot happen.
On average, particles travel short distances {mean free path}| between collisions. Mean free path is collision-frequency inverse and measures average distance between gas molecules. Mean free path decreases as mass decreases, temperature increases, cross-sectional area increases, and density increases.
Systems have different motions and kinetic energies {degrees of freedom, partition}, such as translations, rotations, and vibrations.
translation
All particles can have translations. Average random translational kinetic energy determines temperature.
rotations
Spherically symmetric molecules cannot have net rotational motion. Linear molecules can have one rotational motion state. Two-dimensional molecules can have two rotational motion states. Three-dimensional molecules can have three rotational motion states.
vibrations
Molecules with chemical bonds can have vibration states. Vibrations can involve one bond and be along bond axis. Vibrations can involve two bonds and be across bond axes. Molecule symmetries can cancel vibration states.
partition
Heat can go equally into all available energy states {partition of energy, heat}|. If molecule has more rotation and/or vibration states, raising temperature requires more energy, because some heat does not become average random translation kinetic energy.
partition: heat capacity
Material heat capacity depends on molecular-motion degrees of freedom. Molecules with more rotation and/or vibration states have higher heat capacity.
partition: equipartition
Motion-type average kinetic energies must be the same {equipartition, energy} {energy equipartition} {principle of equipartition of energy}, because energy transfers freely among states by collisions.
amount
Partition average kinetic energy KE is half Boltzmann constant k times temperature T: KE = 0.5 * k * T.
As substance state changes, one mole loses or gains heat {latent heat}|. Latent heat changes molecular conformation and depends on substance type. Total heat Q needed to change state depends on substance mass m and latent heat L: Q = m*L. For liquid-to-gas state change, one mole of liquid gains heat {heat of vaporization} {vaporization heat}. For solid-to-liquid state change, one mole of solid gains heat {heat of fusion} {fusion heat}.
Heating or cooling material can change phase {state change}| {change of state}, by changing chemical arrangement. Molecules spread farther apart, or pack closer together, and change potential energy. Phase changes can happen when increased heat translational kinetic energy causes increased volume, which favors new electrical-attraction structures.
types
State change can be condensation, vaporization, solidification, sublimation, or fusion.
time
Phase change is usually rapid.
temperature
Temperature is constant during state change, because added or removed kinetic energy goes into potential energy change, so average random translational kinetic energy is constant.
pressure
More pressure tends to lower state from gas to liquid to solid, because it compresses molecules and so lowers potential energy.
Adding heat to liquid can increase liquid vaporization until vapor pressure equals air pressure {boiling}|. Heating fluid makes bubbles. Bubbles are liquid vapor, not air bubbles. Boiling is only on pot bottom, because bottom is hottest.
Liquid-to-gas state change is at a temperature {boiling point}| and pressure.
Surface-molecule collisions make some molecules have enough energy to leave surface and make vapor, which has pressure {vapor pressure}|. Molecules that left liquid before can later fall back into liquid from vapor, so vapor pressure depends on outside pressure and temperature. Substances in liquid mixtures contribute partial pressure to total vapor pressure. Total vapor pressure equals sum of partial pressures. Mixed-liquid vapor pressure is less than pure-liquid vapor pressure. Vapor pressure equals mole fraction times pure-vapor pressure: P = f * P0.
State change from liquid to gas is easier if material has weaker bonds between molecules {volatility}|. Materials with small non-polar molecules, globular shape rather than linear shape, and small forces between molecules are volatile. Volatility is high if chemical potential is high. Solute amount that can vaporize depends on boiling point and vaporization enthalpy. If both are low, solute disrupts easily and leaves.
Vapor and liquid {azeotrope}| can have same composition, if they form third material or help each other dissociate.
Partial pressure equals substance mole fraction in liquid times total pressure {Dalton's law} {Dalton law}. Total vapor pressure equals sum of partial pressures.
Liquid can change to gas {vaporization}|. Vaporization causes drying. Liquid-to-gas state change is at a boiling point temperature and pressure. As liquid becomes gas, gas absorbs heat and cools surroundings, as in refrigeration and air conditioning.
Gas can change to liquid {condensation, gas}|. Gas-to-liquid state change is at a temperature {condensation point} and pressure. Cold surfaces cool nearby air and cause air to lose water, which forms surface droplets.
Solid can change to gas {sublimation, heating}|. Solid-gas state change is at a temperature {sublimation point} and pressure.
Solid can change to liquid {fusion, melting}| {melting}. Solid-to-liquid state change is at a temperature {melting point} and pressure.
If pressure on melted ice decreases, ice freezes again {regelation, ice}|.
If liquid has no dirt, bubbles, or other crystallization initiators, it can cool below freezing point {supercooling}| without solidifying.
Liquid or gas can turn into solid {solidification}| {freezing} upon heat loss or removal. Solid-liquid state change is at freezing-point temperature and pressure.
about mixture-solidification temperature {eutectic}.
Physical systems have particles with properties, locations, times, motions, energies, momenta, and relations. Particles can be independent or depend on each other.
degrees of freedom
Related particles have motion restrictions. Particles with no relations are free to move in all directions by translations, vibrations, and rotations. Systems have interchangeable states {degrees of freedom, entropy}. More particles and more particle independence increase degrees of freedom.
order
Order depends on direction constraints. Ordered systems have few possible states. Disordered systems have many possible states. Systems with high heat have more disorder because kinetic energy goes in random directions and potential energy decreases. Systems with work have less disorder because kinetic energy goes in one direction and potential energy increases in direction against field. Systems have disorder amount {entropy, heat}|.
information
Systems with no relations have no information, because particles move freely and randomly, with no dependencies. Systems with relations have information about relations and dependencies. Systems with more degrees of freedom, less order, and more entropy have less information. Systems with fewer degrees of freedom, more order, and less entropy have more information. Because entropy relates to disorder, entropy relates to negative information.
information: amount
The smallest information amount (bit) specifies binary choice: marked or unmarked, yes or no, 0 or 1, or on or off. The smallest system has two possible independent states and one binary choice: 2^1 = 2, where 2 is number of states and 1 is number of choices. Choices can always reduce to binary choices, so base can always be two. Systems have number of binary choices, which is bit number.
information: probability
The smallest system is in one state or the other, and both states are equally probable, so states have probability one-half: 1/2 = 1 / 2^1. State probability is independent-state-number inverse.
information: states
Systems have independent states and dependent states. Dependent states are part of independent states. Systems can only be in one independent state. Particles have free movement, so independent states can interchange freely and are equally probable. Particles have number {degrees of freedom, particle} of independent states available. Systems have number of states. Number is two raised to power. For example, systems can have 2^6 = 64 states. States have probability 1/64 = 1 / 2^6. 6 is number of system information bits. Systems with more states have more bits and lower-probability states.
information: degeneracy
Different degenerate states can have same properties. For example, systems with two particles can have particle energies 0 and 2, 2 and 0, or 1 and 1, all with same total energy.
information: reversibility
Particle physical processes are reversible in time and space. Physical system states can go to other system states, with enough time.
entropy: probability
Disorder depends on information, so entropy depends on information. Entropy is negative base-2 logarithm of probability. For example, for two states, S = -log(1 / 2^1) = +1. For 64 states, S = -log(1 / 2^6), so S = -log(1 / 2^6) = +6. More states make each state less likely, so disorder and entropy increase.
entropy: degeneracy probability
Degenerate-state groups have different probabilities, because groups have different numbers of degenerate states. Groups with more members have higher probability because independent states have equal probability. Entropy depends degeneracy pattern. Going to lower probability group increases system order and has less entropy. Going to higher probability group decreases system order and has more entropy.
Lowest-probability groups are reachable from only one other state. High-probability groups are reachable from most other states. Systems are likely to go to higher-probability groups. Systems move toward highest-probability group. In isolated closed systems, highest-probability group has probability much higher than other groups. If system goes to lower-probability group, it almost instantly goes to higher-probability group, before people can observe entropy decrease. Therefore, entropy tends to increase.
entropy: additive
Entropy and disorder are additive, because they depend on independent states, degrees of freedom. Systems with independent parts have entropy equal to sum of part entropies. If parts are dependent, entropy is less, because number of different states is less.
entropy: heat
Heat is total random translational kinetic energy. Temperature is average random translational kinetic energy. Entropy S is heat energy Q, unavailable to do work, divided by temperature T: S = number of independent particle states = (total random translational kinetic energy) / (average random translational kinetic energy) = Q/T. Kinetic energy is random, and potential energy holds molecules apart in all directions, so heat has no net direction. Average direction is zero.
entropy: energy
At constant pressure and temperature, entropy is enthalpy change divided by temperature, because heat is enthalpy change at constant pressure and temperature. At constant volume and temperature, entropy is energy change divided by temperature, because heat is energy change at constant volume and temperature.
entropy: gravity
If no gravity, entropy increases as particles spread, because particle occupied volume increases. If gravity, entropy increases as particles decrease separation, because potential energy becomes heat though particle occupied volume decreases. If antigravity, entropy increases as particles increase separation, because potential energy becomes heat and particle occupied volume increases.
entropy: mass
Entropy increases when particle number increases. Matter increase makes more entropy. Entropy increases when particles distribute more evenly, toward thermal equilibrium, and have fewer patterns, lines, edges, angles, shapes, and groupings.
entropy: volume
If there are no forces, volume increase makes more possible molecule distributions, less order, and more entropy. If there are forces, volume decrease makes more possible molecule distributions, less order, and more entropy. See Figure 1.
entropy: directions
Energy dispersal increases entropy, because disorder increases. Increasing number of directions or motion types increases entropy. Mixing makes more disorder and more entropy. More randomness makes more entropy. More asymmetry makes more disorder and more entropy.
entropy: heat
Work makes more friction and heat and more entropy. Making heat makes more randomness and more entropy.
entropy: fields
Field strength decrease disperses energy and makes more entropy.
entropy: force and pressure
Pressure decrease disperses energy and makes more entropy. Force decrease disperses energy and makes more entropy.
entropy: volume
At phase changes, pressure change dP divided by temperature change dT equals entropy change dS divided by volume change dV, because energy changes must be equal at equilibrium: dP / dT = dS / dV, so dP * dV = dT * dS. Volume increase greatly increases entropy.
entropy: increase
Systems increase entropy when disorder, degrees of freedom, and disinformation increase. Information decrease makes more interactions and more entropy. Order decrease, as in state change from liquid to gas, increases entropy.
entropy: decrease
Many factors increase order, regularity, or information, such as more regular space or time intervals, as in stripes and waves. Higher energy concentration, more mass, larger size, higher density, more interactions, more relations, smaller distances, closer interactions, more equilibrium, more steady state, more interaction templates, more directed energy, and more filtering increase order.
More reference point changes, more efficient coding or language, better categorization or classification, more repetition, more shape regularity, and more self-similarity at different distance or time scales increase order. More recursion, bigger algorithms, more processes, more geodesic paths, more simplicity, lower mixing, and more purity increase order. More reconfigurations, more object exchanges, and more combining systems increase order. Fewer functions, fewer behaviors, more resonance, fewer observations, more symmetry, more coordinated motion, and more process coupling increase order.
Larger increase in potential energy increases order, because energy concentrates. Higher increase in fields increases order, because energy increases. Fewer motion degrees of freedom, as in slow and large objects, increase order. More same-type, same-range, and same-size interactions increase order. More and equal influence spheres increase order. Higher space-time curvature increases order. More constant space-time curvature increases order.
Lower harmonics of Fourier series increase order. Fewer elements in Fourier series increase order.
entropy: closed system
In closed systems, entropy tends to increase, because energy becomes more random. Potential energy becomes random kinetic energy by friction or forced motion. Random kinetic energy cannot all go back to potential energy because potential energy has direction. Work kinetic energy becomes random kinetic energy by friction or forced motion. Random kinetic energy cannot all go back to work energy because work energy has direction. Heat energy is already random. Only part, in a direction, can become potential energy or work kinetic energy. Radiation becomes random kinetic energy by collision. Random kinetic energy cannot all go back to radiation energy, because radiation energy requires particles accelerated in direction.
universe entropy
Universe is isolated closed system. It started in low-entropy state and moves to higher entropy states.
Perhaps, at beginning {hot big bang} {primordial fireball}, universe had one particle at one point with smallest possible volume, and so no relations among parts. There were no space fields and no tidal effects. Universe had highly concentrated energy at high temperature and so large contracting forces and high pressure. Particle number remained the same or increased, as particles and radiation split.
Universe expansion increased space volume. Space points became farther from other points. Expansion was greatest at first. Then expansion slowed, because all particles had gravity.
As universe cooled, it created particles, in evenly distributed gas. Entropy increased but was still low.
As universe cooled, gas-particle gravity formed galaxies and stars. Condensed gas had higher entropy but was still low. Potential energy converted to heat as infalling particles collided. Heat and mass concentrated in stars.
Stars are hot compared to space, so stars can transfer energy to planets and organisms. Stars undergoing nuclear fusion make visible light. Visible light has higher energy than heat infrared radiation. On Earth, temperature stays approximately constant. Therefore, visible light energy that impinges on Earth is equal to energy that radiates away from Earth as heat. Because sunlight has higher energy per photon, fewer sunlight photons land on Earth than Earth emits as infrared heat photons. Entropy increases in space, and total universe entropy increases. On Earth, order increases and disorder decreases, mostly in organisms. From universe beginning until now, universe entropy increases, while small-region physical forces and particle motions can cause entropy decreases.
Now, universe has many photons, large volume, negligible forces, and even matter distribution, so universe entropy is now large. For example, cosmic microwave background radiation has many randomly moving photons, from soon after universe origin. Photons mostly evenly distribute. They fill whole universe and have little effect on each other. As universe expands, their entropy becomes more.
Now, universe has many galaxies with central black holes and has black holes formed after supernovae. Black holes are mass concentrations denser than atomic nuclei. Black holes have very high entropy, because particle number is high, volume is small, mass evenly distributes, gravitational force and fields are high, and density is high. Black holes make universe entropy large now.
In the future, universe entropy will increase. Universe will have more black holes and can evolve to have only black holes. Universe will have more local forces. Universe will have more volume. Universe will have more particles. Universe will have more-even particle distribution.
Systems have particles, which have position and momentum, and energy and time {information, physics}.
spin system
Particles can have spin up clockwise or spin down counterclockwise. Particle spin state encodes one binary choice or information bit: 0 or 1. For quanta, such as spin, state holds one quantum information bit. One kilogram of plasma in one liter of space can hold 10^31 bits.
Particles with known spins can carry input or output information. For interacting particles, quanta can entangle, and information can entangle. Particles can interact to represent computation to calculate. Bits can change electromagnetically every 10^-20 second. Signals travel 3 * 10^-12 seconds to next particle. System particles entangle and change in parallel. One kilogram of plasma in one liter of space can have 10^51 particle-spin changes each second.
system information
Information required to describe system state depends on degrees of freedom. Mole of chemical has 10^23 degrees of freedom because it has that many molecules. In regions that have no boundary and have uniform matter and energy, entropy is proportional to volume.
black hole
By Bekenstein-Hawking formula, one-kilogram black holes have radius 10^-27 meters. Event-horizon surface can hold 10^16 bits. Bits can change every 10^-35 second. Surface can have 10^51 particle-state changes each second. Light can cross black holes in 10^-35 seconds, so physical processes have time to work serially.
Observers cannot see past event horizon, so horizon surface must contain all information about region volume inside black hole. Gravity causes information bits to interact, so entanglement over surface area can have enough information states to hold information about inside volume. Black-hole entropy is directly proportional to event-horizon surface area. Planck area is 10^-66 cm^2. One Planck area has 0.25 information bit, so four Planck areas make one bit.
One-kilogram black holes emit gamma and higher-energy rays by Hawking radiation and disappear in 10^-21 second. Smaller black holes dissipate faster and emit higher energies.
Perhaps, rather than particle spins and states, branes or strings hold states.
cell size
Bit-change maximum rate is maximum clock frequency, and so time has quanta. Rate depends on gravitational constant, light speed, and Planck constant.
Space has quanta, with cell sizes. Black holes have maximum energy per volume and have definite ratio between mass and event-horizon radius, so maximum mass-energy is proportional to space-time radius. Cell size varies with radius cube root. For universe, cell diameter is 10^-15 meters.
Energy and time relate, so space cell size depends on space-time radius. The holographic principle can derive from uncertainty principle.
universe
Universe has age 1.3 * 10^10 years and so radius 1.3 * 10^10 light-years. Universe is close to maximum density, so it can hold 10^123 bits. It can have 10^106 bit changes each second, in parallel, from 10^72 joules. It has 10^92 bits of ordinary matter, which can change bits at 10^14 Hz. It has less than 10^123 bits of dark energy, which can change bits at 10^18 Hz.
Total entropy in universe, inside and outside black holes, cannot go down {generalized second law} (GSL).
Information that went into black hole can come out by entanglement of two surface virtual particles, one of which interacts with original matter and information at singularity {Horowitz-Maldacena model}.
Information bits can change no faster than quantum time. Heisenberg uncertainty principle relates energy and time. Quantum time t depends on quantum energy E {Margolus-Levitin theorem}: t >= h / (4*E), where h = Planck's constant.
Masses in volumes can hold limited entropy amounts {universal entropy bound}.
Energy and mass within volume can hold limited entropy amount {holographic bound}. Holographic bound is number of event-horizon-surface Plank areas divided by four. Maximum-entropy regions relate mass and volume, so area can define region information. Black-hole formation prevents volume entropy increases from overtaking surface entropy increases and so limits entropy.
Particle systems have total energy and distribute energy among particles {statistical mechanics}|.
energy: particle
Particles have energy levels. Particles have possible energy levels. Particle energy level cannot be zero, because particles must move and so have kinetic energy. Particles always have at least minimum ground-state energy, because energy has quanta.
energy: distribution
Some particles have lower energy, and some have higher energy {distribution, energy}. Particles exchange energy by collisions or electronic transitions. Systems have average particle energy, which is higher for higher temperature and/or work. Large systems typically have only one particle-energy distribution, which has highest probability.
energy: total
Sum of particle energies equals total energy. Total energy equals average particle energy times particle number.
energy: types
Particles can have translational energy, vibrational energy, rotational energy, and electronic-transition energy, with different ground states and different quanta. At normal temperatures, vibrational energy is at ground state, electronic-transition energy is at ground state, and rotational energy is above ground state. Total energy distributes equally among possible translation, rotation, vibration, and electronic-transition energy levels, if there are pathways. Systems with large energy quanta have few particles at high-level energies. Systems with small energy quanta have more particles at high-level energies. Energy change does not change particle distribution much.
entropy
Energy distributions have entropy. Entropy change changes particle distribution. Systems with few particles or low temperatures have quantum states, easy transitions among states, and minimal entropy. Systems with many particles or high temperature have thermal states. Black-hole event horizons have random kinetic energy and cause thermal states. Thermal states have random kinetic energy and have maximum entropy.
entropy: degeneracy
Different particle-energy distributions can have same number of particles at each energy level. For example, if two same-type particles exchange energies, system has different particle-energy distribution, same total energy, and same number of particles at each energy level. Different particle-energy distributions with the same energy and same number of particles at each energy level make system phase. System has largest phase, which has highest probability and most even energy distribution possible at total energy. Largest phase has highest entropy.
state: fluctuation
If system is in largest phase, particles have lowest probability of returning to smaller regions, because largest phase has highest probability. If particle collision results in smaller phase, in shortest possible time, system returns to largest phase, because largest phase has highest probability. Hawking radiation requires large phase fluctuation.
For systems with many molecules at equilibrium at temperature, frequency distributions {Boltzmann distribution, statistics}| can plot frequency against molecule energy. y(E) = e^(- E / (k * T)), where E is molecule energy level, y(E) is frequency for molecule energy level, e is natural-logarithm base, k is Boltzmann constant, and T is absolute temperature.
energy
Particle-energy probability is partition number and is relative frequency of that energy in Boltzmann distribution. Most-probable energies are near average energy. Total energy is integral of Boltzmann distribution.
comparison
At temperatures above 50 K, Boltzmann distributions look like Gaussian distributions.
equilibrium
Systems at equilibrium have Boltzmann distribution, because that distribution has much higher probability than other distributions with same total energy. Boltzmann distribution has the most combinations that can give total energy.
equilibrium: entropy
For that reason, Boltzmann distribution has lowest probability of molecule being in any one energy level, so Boltzmann distribution has the most entropy and least order. Entropy S equals Boltzmann constant k times combination-number C natural logarithm: S = k * ln(C).
Molecular properties {canonical properties} can be at constant temperature {canonical property}, at constant temperature and volume {grand canonical property}, or in isolated adiabatic systems {microcanonical property}.
If system molecules are indistinguishable, some particle-energy distributions have same numbers of particles at each energy level {degeneracy, system}|.
Particles have different possible motions and kinetic energies {degrees of freedom, energy}.
Physical systems can have different numbers and energy levels of particles {distribution of energies} {energy distribution}. Particles can be molecules, atoms, photons, or subatomic particles.
energy quanta
Particle energy cannot be zero, because particles are always moving and so have kinetic energy. Particle energy has quanta, by quantum mechanics, so particles have lowest energy level {ground-state energy}. Particle energies increase from ground-state energy by discrete energy quanta. Possible particle energies are ground-state energy, ground-state energy plus one quantum, ground-state energy plus two quanta, and so on. For total energy, possible energy levels have numbers of particles. Systems have particles at ground-state energy, particles at ground-state energy plus one quantum, particles at ground-state energy plus two quanta, and so on. Particle number at high energy levels is small compared to number at low energy levels, because elastic collisions distribute energy among energy levels. High particle energy has low probability. Infinite particle energy has zero probability.
system energy
Closed systems have constant total energy. Total energy is ground-state energy times particle number, plus any quanta. Sum of particle energies makes total energy. Product of particle number and ground-state energy is minimum system energy.
energy distribution
For example, two-particle system can have one particle with energy 3, one particle with energy 1, and total energy 4. For closed systems, particle collisions can change energy distribution, but total energy stays constant. For example, the two-particle system can have one particle with energy 2, one particle with energy 2, and total energy 4.
energy distribution: low-energy example
Two-particle system can have ground-state energy Q0, one particle at ground-state energy, E1 = Q0, and another particle at one quantum energy level Q above ground-state energy, E2 = Q0 + 1*Q. Total energy is E1 + E2 = Q0 + (Q0 + 1*Q) = 2*Q0 + 1*Q. See Figure 1.
energy distribution: equivalent distributions
For closed systems, different energy distributions can result in same total energy. For example, twelve-molecule systems can have energy distributions in which each particle has energy Q1a and total energy is 12*Q1a. By particle collision, system can have energy distribution with six molecules one quantum Q above Q1a and six molecules one quantum Q below Q1a. System still has total energy 12*Q1a.
For two-molecule system with total energy 2*Q0 + 2*Q, both molecules can have energy Q0 + 1*Q. After collisions, first molecule can have energy Q0, and second molecule can have energy Q0 + 2*Q, or first molecule can have energy Q0 + 2*Q, and second molecule can have energy Q0. See Figure 2. All three energy distributions have same total energy.
probability
In closed physical system, all energy distributions have same total energy, and all distributions are equally likely, because collisions transfer energy freely between particles.
probability: distinguishable particles
If particles are distinguishable, energy distributions are unique and have equal probability. For example, system can have total energy 6, ground-state energy 2, quantum 1, and 2 particles. If particles are distinguishable, such as E1 and E2, three energy distributions are possible. E1 = 2 and E2 = 4. E1 = 4 and E2 = 2. E1 = 3 and E2 = 3. Distribution E1 = 3 and E2 = 3 has one-third probability. Distribution E1 = 2 and E2 = 4 has one-third probability. Distribution E1 = 4 and E2 = 2 has one-third probability. Distributions are equally likely. See Figure 3.
In this system, particles cannot have energy 0 or 1, because ground-state energy is 2. Only these three cases make total energy 6.
probability: indistinguishable particles
Typically, some system particles are exactly the same and so indistinguishable. For example, all electrons are the same. If particles are indistinguishable, some energy distributions appear the same.
For example, system can have total energy 6, ground-state energy 2, quantum 1, and 2 indistinguishable particles. Two energy distributions are possible: energy 2 and energy 4 or energy 3 and energy 3. Cases E1 = 2 and E2 = 4, and E1 = 4 and E2 = 2, are now indistinguishable. Energy distribution 3 and 3 happens once and has one-third probability. Energy distribution 2 and 4 happens twice and has two-thirds probability.
probability: degeneracy
If particles are indistinguishable, some energy distributions have same numbers of particles at each energy level. In the example, two energy distributions have one particle at level 2 and one particle at level 4, so degeneracy is two.
Degenerate energy-distribution probability is degeneracy divided by number of distributions when particles are distinguishable. In the example, number of energy distributions with distinguishable particles is three. Degeneracy of "energy 2/energy 4" distribution is two, and probability is 2/3. Degeneracy of "energy 3/energy 3" distribution is one, and probability is 1/3.
For degenerate distributions, more degeneracy makes higher probability. Degeneracy is greater if most particles are near average energy and particles have Boltzmann energy distribution. Maximum degeneracy spreads particles maximally. For many-particle systems, highest probability is many orders of magnitude above second-most-likely distribution.
partition number
If system has constant total energy, distribution degeneracy {partition number, distribution} is (total number of particles)! / (number at ground-state energy)! * (number at ground-state energy plus one quantum)! * ... * (number at ground-state energy plus infinite number of quanta)!, where ! means factorial. Above 50 K, for thermal distributions, partition number maximizes according to Boltzmann distribution.
probability: particle
Particle has probability that it is in energy level. If system is in energy distribution with maximum degeneracy, particle has lowest average probability that it is in energy level, because particles spread most evenly. Other energy distributions increase average probability that particle is in energy level, because they concentrate particles more.
system state
With collisions, systems tend to go to most degenerate energy distribution, from which the most collisions make same degenerate distribution, because it repeats itself the most. This is why the most-degenerate energy distribution has highest probability. Isolated systems soon reach this single stable state and stay there.
Complex systems can have sets {ensemble, system}| of identical objects. Sets have statistical properties. Linear ensemble operators can calculate set-property average values, while varying initial conditions.
Collisions interchange energy, so average energies of system kinetic-energy sources are equal {equipartition, statistical mechanics}|.
motions
For particles, kinetic energy partitions equally into available motion states {degrees of freedom, motion}. Particles have different possible motions. Translations can independently be in three spatial dimensions. Vibrations depend on chemical-bond stretching-and-bending modes. Rotations can be in zero to three rotation dimensions, depending on molecule symmetry.
energy
If temperature is above 50 K, average partition energy E is half Boltzmann constant k times absolute temperature T: E = 0.5 * k * T. System partition function is product of particle partition functions.
In rare systems, all phases solve energy equation, and system reaches equilibrium {ergodic hypothesis}|. Though few systems are ergodic, real systems come arbitrarily close to ergodic {quasiergodic hypothesis}. Quasiergodic systems are fractal, because one trajectory cannot fill up space but can pass close to all points. If trajectory does not follow simple law, system uses statistical law.
Plane rectangles can have many square cells and some particles {gas in box model}. Connected cells make a region with percentage of total cell number. For example, box can have 10 cells, with one-cell region in one corner. Probability that one particle is in region is 1/10.
Statistical thermodynamics applies to systems {ideal fluid} in which the only interactions among particles are elastic collisions, with no forces between molecules. Particles can have cross-sections.
Boltzmann distribution gives number of molecules at each energy level {partition function}| {canonical partition function}, for a temperature.
Probability {partition number, energy} of particle energy level is relative frequency of that energy in Boltzmann distribution. Most-probable energies are near average energy.
Molecule collisions make fast and discrete molecule-energy changes {quantum energy change}.
temperature
Energy fluctuations Q depend on Boltzmann constant k times absolute temperature T: Q = k*T. At higher temperatures, energy change is more, and molecule energy levels are farther apart. At low temperature, quanta are almost equal, and molecules have ground-state energy Q0 plus multiple of energy quantum Q: Q0, Q0 + 1*Q, Q0 + 2*Q, Q0 + 3*Q, and so on.
factors
Quantum increases if volume decreases, system does work, mass decreases, temperature decreases, pressure decreases, fields decrease, or electrons transition to lower orbits. In those cases, overall energy decreases, so quanta are bigger.
entropy
If system has only entropy changes, quanta stay the same.
high energy
If quanta are large, high energies are hard to reach.
Spontaneous processes {spontaneous process}| lower free energy. Particles move along geodesics. Electrons move along zero-field lines. Particles orbit at lowest orbit.
Coldness can be very low {cryogenics}|. Lasers cool by slowing atoms to 50 microKelvin. Magnetic fields can trap and compress gas. Cooling can be by both lasers and magnetic traps {magneto-optical trap} (MOT).
Evaporation cools to below 50 microKelvin by removing hottest atoms. Time-averaged orbital potential (TOP) magnetic trap allows evaporation at point that moves in circle to build gas ellipsoid. Ioffe trap holds plasmas. Ioffe-Pritchard trap can use parallel magnetic fields or other arrangements to form various gas shapes.
properties
Quantum-mechanical effects can change low-temperature material properties, such as superconductivity.
Metal superconductors have bound-electron pairs, each with same spin, which make metal ions, streamline flow, and make bosons that can condense {Bose-Einstein condensation, cryogenics} (BEC). All bosons in same quantum state can condense from gas to make liquid. Repulsive bosons condense better.
At low temperature, fluids {superfluid}| can have no viscous resistance. Liquid helium is the only known superfluid. Vortexes but no overall rotation can appear in spun superfluids. Space-time can be like fluid, and black-hole event horizon can be like superfluid with quantum-phase transition. General relativity has same equations as sound waves in moving fluid.
Helium 4 can cool and compress to solid {supersolid}, with no viscous resistance.
High temperature can separate electrons from atoms and cause electrons to leave metal or metal-oxide surface {Edison effect} {thermionic emission, heat}|.
Thermionic emission leaves surface positive charge {space charge, thermionic}.
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Date Modified: 2022.0225