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Perhaps what makes the second law so remarkable is that it describes ''irreversible'' phenomena.  In particular, it describes the observed fact that heat energy, in bodies that are not being externally manipulated by compression, etc., flows only from a warmer body to a cooler one.  When a warmer body is placed in contact with a cooler one, heat energy will flow (always perserving total energy, of course) from the warmer one to the cooler one.  The warmer one will cool off as it releases its energy, and the cooler one will warm up.  This process will continue until the two bodies reach the same temperature, or "thermal equilibrium".

Perhaps what makes the second law so remarkable is that it describes ''irreversible'' phenomena.  In particular, it describes the observed fact that heat energy, in bodies that are not being externally manipulated by compression, etc., flows only from a warmer body to a cooler one.  When a warmer body is placed in contact with a cooler one, heat energy will flow (always perserving total energy, of course) from the warmer one to the cooler one.  The warmer one will cool off as it releases its energy, and the cooler one will warm up.  This process will continue until the two bodies reach the same temperature, or "thermal equilibrium".

Until the development of statistical mechanics, no one knew why this was so, or what temperature actually meant.  What was known was simply that a body with a higher temperature would send heat to a body with a lower temperature, no matter what the bodies were made of.

Until the development of [[statistical mechanics]], no one knew why this was so, or what temperature actually meant.  What was known was simply that a body with a higher temperature would send heat to a body with a lower temperature, no matter what the bodies were made of.

The fact that this kind of heat flow is irreversible makes the whole field of thermodyamics lie outside of the realm of classical Newtonian mechanics or Relativistic mechanics.  In Newtonian or Relativistic mechanics, every phenomenon can go in reverse order.  The catchy phrase "arrow of time" (or "time's arrow") was coined by [[Arthur Eddington]] to denote this one-way behavior not shared by other theories of physics.<ref>''The Nature of the Physical World'', Arthur Eddington, MacMillan, 1929, ISBN 0-8414-3885-4</ref>

The fact that this kind of heat flow is irreversible makes the whole field of thermodyamics lie outside of the realm of classical Newtonian mechanics or Relativistic mechanics.  In Newtonian or Relativistic mechanics, every phenomenon can go in reverse order.  The catchy phrase "arrow of time" (or "time's arrow") was coined by [[Arthur Eddington]] to denote this one-way behavior not shared by other theories of physics.<ref>''The Nature of the Physical World'', Arthur Eddington, MacMillan, 1929, ISBN 0-8414-3885-4</ref>

Prior to the insight of [[quantum mechanics]], which established the fundamental underlying uncertainty in the universe, the field of statistical mechanics attributed the increase in entropy to the statistical tendencies of huge aggregates of particles at the molecular or atomic level.  While Newtonian and Relativistic mechanics can, in principle, precisely describe assemblages of any number of particles, in practice they are not directly applied to the behavior of bulk material.  That is, they are not applied to a number of particles on the order of Avogadro's number.

The field of statistical mechanics attributes the increase in entropy to the statistical tendencies of huge aggregates of particles at the molecular or atomic level.  While Newtonian and Relativistic mechanics can, in principle, precisely describe assemblages of any number of particles, in practice they are not directly applied to the behavior of bulk material.  That is, they are not applied to a number of particles on the order of Avogadro's number.

Statistical mechanics was developed in the 19^th century prior to quantum mechanics, so rather than attributing the Second Law to the fundamental uncertainty in nature, statistical mechanics bases its models on assumptions concerning the statistical behavior of large numbers of particles.

Statistical mechanics makes no assumptions about the microscopic cause of the seemingly random behavior.  Statistical mechanics was developed in the 19^th century prior to quantum mechanics, and does not depend on the [[Heisenberg uncertainty principle]].  The quantum mechanical uncertainty goes away at the microscopic level once the system is observed and the wave function collapses.

==Probability and statistics==

 

==Elementary probability and statistics==

If you flipped a coin 20 times and it came up heads each time, you would consider that to be a remarkable occurrence.  (Perhaps so much so that you would inspect the coin to be sure it didn't have heads on both sides.)  If you tried it and got tttthtththhhththhthh, you would probably not consider it remarkable.  Yet each of these outcomes is equally probable: about 1 in 10⁶.  If you shuffled a deck of cards and found them all in exact order from the 2 of clubs to the ace of spades, you would consider that to be very remarkable.  But if you got the distribution shown in the illustration on page 314 of Alfred Sheinwold's ''5 Weeks to Winning Bridge'', you would probably consider it just "random".  Yet each of these orderings has the same probability of occurring&mdash;1 in 52 factorial, which is about 10⁶⁶.<ref>Actually, we're ignoring the fact that bridge players sort their cards by suit, and the diagram shows the result of the sorting.</ref>

If you flipped a coin 20 times and it came up heads each time, you would consider that to be a remarkable occurrence.  (Perhaps so much so that you would inspect the coin to be sure it didn't have heads on both sides.)  If you tried it and got tttthtththhhththhthh, you would probably not consider it remarkable.  Yet each of these outcomes is equally probable: about 1 in 10⁶.  If you shuffled a deck of cards and found them all in exact order from the 2 of clubs to the ace of spades, you would consider that to be very remarkable.  But if you got the distribution shown in the illustration on page 314 of Alfred Sheinwold's ''5 Weeks to Winning Bridge'', you would probably consider it just "random".  Yet each of these orderings has the same probability of occurring&mdash;1 in 52 factorial, which is about 10⁶⁶.<ref>Actually, we're ignoring the fact that bridge players sort their cards by suit, and the diagram shows the result of the sorting.</ref>

With really large numbers, the probability of any particular outcome is vanishingly small; the only sensible measure is the accumulated probability, or the probability density, measured in a way that doesn't involve individual outcomes.

With really large numbers, the probability of any particular outcome is vanishingly small; the only sensible measure is the accumulated probability, or the probability density, measured in a way that doesn't involve individual outcomes.

When dealing with thermodynamics, we are dealing with the statistical aggregate behavior of macroscopic pieces of matter, so we have to increase the number of items from 10, or 52, to something like [[Avogadro's number]].  So the number of possible situations, instead of being 10⁶ or 10⁶⁶, is something like 10^{Avogadro's number}, that is, 10^10²³.  The enormity of such a number makes a huge amount of difference.

 

 

 

When dealing with thermodynamics, we are dealing with the statistical aggregate behavior of macroscopic pieces of matter, so we have to increase the number of items from 10, or 52, to something like [[Avogadro's number]].  So the number of possible situations, instead of being 10⁶ or 10⁶⁶, is something like 10^{Avogadro's number}, that is, 10^10²³.  The enormity of such a number makes a huge amount of difference. If you flip a coin Avogadro's number of times, it will come up heads about half the time, as before.  But, for all practical purposes, we can say that it will come up heads ''exactly'' half the time.  The number of heads might be off by a few quintillion (this is the "law of large numbers"), but that won't make any practical difference.

 

*You can't ask any questions about individual items&mdash;air molecules don't have labels like "Jack of Diamonds".  You can only ask questions about the aggregate behavior of macroscopic pieces of space.

*You can't ask any questions about individual items&mdash;air molecules don't have labels like "Jack of Diamonds".  You can only ask questions about the aggregate behavior of macroscopic pieces of space.

*While the probabilities of certain outcomes can be mathematically calculated, they are so small that, as a practical matter, we can say that '''they do not occur'''.  People sometimes like to say things like "The second law of thermodynamics means that it is very unlikely that heat will travel from a colder object to a warmer one."  That's a fallacious way of thinking about it.  It is a [[statistical impossibility]]&mdash;it just doesn't occur.

*While the probabilities of certain outcomes can be mathematically calculated, they are so small that, as a practical matter, we can say that '''they do not occur'''.  People sometimes like to say things like "The second law of thermodynamics means that it is very unlikely that heat will travel from a colder object to a warmer one."  That's a fallacious way of thinking about it.  It is a [[statistical impossibility]]&mdash;it just doesn't occur.

==Application to molecular behavior==

==Application to molecular behavior==

The development of the kinetic theory of gases, statistical mechanics, and thermodynamics revolutionized 19^th century physics.  It was recognized that, while we can't analyze the behavior of every molecule, we can analyze the statistical behavior of macroscopic assemblages.  When gas molecules collide, they can transfer energy in a manner that leads to the principle of ''equipartition of energy''.  This, plus the constraints on conservation of the total energy, leads to the ''Maxwell-Boltzmann'' distribution of molecular energies.  From this, one can deduce the properties of volume, pressure, and temperature, leading to Boyle's law and Charles' law, among others.  Temperature was found to be just the average energy per molecule.  (Actually, the average energy per "degree of freedom".)  The equipartition principle was found to give an explanation of the fact that, when two bodies are in contact, they exchange energy in a way that makes the warmer body get cooler and the cooler body get warmer, always in accordance with conservation of energy.

 

[[Statistical mechanics]] is the application of probability to enormous numbers like this.  It was developed, along with the [[kinetic theory]], by Maxwell, Boltzmann, Causius, Clapeyron, and others, during the 19^th century.  The development of the kinetic theory of gases, statistical mechanics, and thermodynamics revolutionized 19^th century physics.  It was recognized that, while we can't analyze the behavior of every molecule, we can analyze the statistical behavior of macroscopic assemblages.  When gas molecules collide, they can transfer energy in a manner that leads to the principle of ''equipartition of energy''.  This, plus the constraints on conservation of the total energy, leads to the ''Maxwell-Boltzmann'' distribution of molecular energies.  From this, one can deduce the properties of volume, pressure, and temperature, leading to Boyle's law and Charles' law, among others.  Temperature was found to be just the average energy per molecule.  (Actually, the average energy per "degree of freedom".)   

 

The fact that heat only flows downhill, and that entropy never decreases, is now just a consequence of the "most probable distribution" principle, or equipartition principle, from mathematical statistics, albeit at a vastly larger scale.

===Example of the Statistical nature of the Second Law===

===Example of the Statistical nature of the Second Law===

Note that the Boltzmann constant has taken a value of 1 to simplify the maths.

Note that the Boltzmann constant has taken a value of 1 to simplify the maths.

So the state that we would expect to find the system in, the last one, has the highest entropy. However, the system could be in this state (10 in left, 10 in the right) and, just by chance, all the molecules could make their way to the left hand side of the box. This corresponds to a '''decrease''' of entropy. This example could be expanded up to a room, so why do we never see all the air in a room suddenly move to one end? The reason is that it is so unlikely, perhaps less than <math>10^{-10^{26}}</math>, than it practically never occurs<ref>{{cite book

So the state that we would expect to find the system in, the last one, has the highest entropy. However, the system could be in this state (10 in left, 10 in the right) and, just by chance, all the molecules could make their way to the left hand side of the box. This corresponds to a '''decrease''' of entropy. This example could be expanded up to a room, so why do we never see all the air in a room suddenly move to one end? The reason is that it is so unlikely, perhaps less than <math>10^{-10^{26}}</math>, that it essentially never occurs<ref>{{cite book

|author=Hugh D. Young and Roger A. Freedman

|author=Hugh D. Young and Roger A. Freedman

|title=University Physics with Modern Physics

|title=University Physics with Modern Physics