Information Theory and Thermodynamics

In puzzle outation Theory and ThermodynamicsIn order to develop better tools, machines and technology we suck had to develop our mind of the physical world. This has whollyowed us to construct machines that atomic second 18 more capable than those former it.The French scientist Carnot was studying machines and was trying to understand how to make them better and more efficient. As part of his studies he calculated the maximum efficiency of either machine and was able to relate this to temperature. Carnots idea was to simplify the machine to its simplest form (this generality that makes it universal) and analyse that Carnot knew that machines of the age (and of today) work as a upshot of a temperature unlikeness across the machine. In his time it was obvious, fire produced go which turned a turbine that did work today is not too different, all told of our machines still need a temperature difference to make them work, however the temperature difference driving the machin es may be at some distance, for good example a power station producing electricity. Even wind, and solar require temperature differences to work.Analysing these machines further train to concepts that we, perhaps, take for granted work, power and energy notable examples. Whilst working on these concepts Boltzmann came up ideas that grew into statistical thermodynamics. It was extended and correctly describes a whole lay out of phenomena.The idea of thermodynamics is to relate various physical properties of a substance to the pouch behaviour of the constituent parts within.Micro evokes and Macro farmingsBoltzmann realised that by k straighta focussinging the subjugate of different states that a arrangement could be in and the number of configurations that individually state would enable him to work out the chance of a situation state returnring. And that on average when something is observed it is more plausibly to be found in angiotensin-converting enzyme of its more pro bable states. Many transcriptions create lots of moving components and this means that over time a system will have evolved into a more probable state. This may now seem obvious, but it hadnt been pointed out explicitly at the time.A Macrostate is the worldwide state of a system. For example if we consider a box with release, unconsolable and green lubbers a possible macrostate cogency be to find all of the red balls atomic number 18 in the bottom left corner, whilst all of the others atomic number 18 randomly distributed in the rest of the box. Another example of a macrostate might be that the integrality electrical charge of fend off could be Coulombs.A Microstate is one particular configuration of the system that produces a macrostate. In the balls example if the balls be identical apart from colour we can permute the balls with the similar colour amongst themselves and end up with different microstates. An example with the charged block might be that we have 4 fra ctions each with charge as one microstate, and another might be to have 1 particle with and another with Below is a figure that represents a hypothetical macrostate of triad colours of balls three particular microstates that can be used to get through it. In the left roughly diagram we have a macrostate with all of the red balls in the bottom left corner, the other diagrams show different permutations of the balls that in any case achieve the desired macrostate.In order to calculate the probability of this state we would need to know how many combinations of it there ar. This is a simple tally argument we have 1 way of putting the green ball in its spot, cardinal ways of putting the blue balls in their range and ways of arranging (we can pick any of the 5 to go in the corner, thence any of the remaining 4 to go contiguous to that, then any of the remaining 3 etc). in that respect are a total of 8 balls and so theIn general if there are objects we have possible arrange ments if we also have different grammatical cases and if of them are of lawsuit 1 (say red), are of type 2 and are of type then we can find the total number of permuted arrangements withWe can use these two facts to calculate the number of accessible microstates of type , this is called the saddle of the microstate and is denoted by, The metric weight unit of a microstate is proportional to the probability of the system being in it. So one way to calculate the probability of being in the state is viawhere the centre is over the weights of all the other possible microstates.A handy way to view a microstate is with a pack of cards (Birks bath), in a pack of playing cards the statistical weight of a hostelry is 13 since there are 13 of then the statistical weight of a queen is 4. The probability of selecting a club card is the chances of pick out out a club are times greater than option out a queen. The statistical weight of the queen of hearts is 1.There is one obviou s constraint that can always be obligate and that is that the total number of particles is the sum up of the number of particles in each stateWe can impose other constraints on the system as they are required later. Because the particles of each type are identical it is natural to narrow down a probability that a randomly selected particle is of type, , asWe are also able to define an Expectation value for the system. If we were interested in the average military control of each of the types we would havewhich would represent the average business of each type. If we were interested in the charge (or energy (I shall use for either) we would as well haveLet us take some examples and compute the statistical weights, average line of work and average energy (represented by the value of the type index e.g. if , the energy would be two units). I shall consider that the atoms are all identical apart from the energy that they have and that a macrostate is the similar for each. For the first case let us assume that we have 5 atoms and the macrostate means to an energy of 5 units. The table infra shows that (for example) the microstate 3 has a weight of 20, this means that there are 20 microstates with the occupancy levels given that correspond to the macrostate We can tabulate the various combinations as belowmicrostate number occupancy of type iweightprobabilityn0n1n2n3n4n5140000150.03972310010200.15873301100200.15874220100300.23815212000300.23816131000200.1587705000010.0079totals151242111261Average occupancy2.2221.3890.7940.3970.1590.040 table 3.1 Table showing occupancy levels for a 5 atom system with a macrostate of 5.This table was generated by finding all of the numbers that sum (in this case) to 5 which is the macrostate. It shows the number of atoms with a particular energy in the columns headed , the statistical weight of each microstate is in the weight column, the probability column bordering to it shows the probability of randomly selecting th is microstate from a given macrostate (in this case 5 atoms and a total energy of 5). The row titled average occupancy shows the evaluate occupancy of an energy level of type , calculated from the table. Looking at the table there are two equally most likely microstate arrangements. The first of these corresponds to and , both occurring with a probability of 0.238.Another possible macrostate is listed below, this time we have 7 atoms and an energy of 7 units. The headings of the table are the aforementioned(prenominal) as in the previous example. We can see that the weight of the most probable microstate is 420 and that we have a probability of 0.245 of randomly selecting one of them. The occupancy levels aremicrostateoccupancy of type iweightprobabilityn0n1n2n3n4n5n6n716000000170.004251000010420.024350100100420.024450011000420.024542000100 one hundred five0.0616411010002100.1227410200001050.0618402100001050.0619330010001400.08210321100004200.24511313000001400.08212240100001050.0 6113232000002100.1221415100000420.024150700000010.001totals5130116321117161Average occupancy3.2311.8851.0280.5140.2280.0860.0240.004Table A3 2 a vii atom system with a total energy of sevenA final example consists of a system of 10 atoms and a total energy of 9. As will be readily seen as the number of atoms and the energy increases the number of microstates corresponding to a given macrostate increases so does the size of the table. It was quite difficult to work out the number of combinations of energy that could occur and I wouldnt want to do it again for larger tables. In the future(a) part we shall use the method of Lagrange multipliers to massively simplify the calculations for the probabilities and expectations. For the case of 10 atoms and an energy of 9 units.We see that the most probable microstates have the succeeding(a) occupancy levelsThe most probable microstate has a probability of 0.1555, but there is another microstate that is only slightly less probable (a prob ability of 0.1300) and this has occupancy levels ofThe two least likely microstates are the followingBoth have a probability of 0.0002 which is very small indeed. Table 3 is belowdoccupancy of each type iweightprobabilityn0n1n2n3n4n5n6n7n8n919000000001100.00020567728100000010900.0018510938010000100900.0018510948001001000900.0018510958000110000900.00185109672000001003600.00740436771100010007200.014808721871010100007200.014808721971002000003600.007404361070200100003600.007404361170111000007200.0148087211270030000001200.002468121363000010008400.01727684114621001000025200.05183052215620110000025200.05183052216612010000025200.05183052217611200000025200.0518305221860310000008400.01727684119540001000012600.02591526120531010000050400.10366104521530200000025200.05183052222522100000075600.15549156723514000000012600.02591526124450010000012600.02591526125441100000063000.12957630626433000000042000.0863842042736010000008400.01727684128352000000025200.0518305222927100000003600.0074043630