Introduction To EntropyThe Way Of The World – Jonathan Allday

Our volume of phase space, in units of , determines the entropy. However, we need to be a little more formal about how we define regions of phase space in the first place. If we gently wave our hands and say15 “ is the region of phase space occupied by reasonably probable microstates that are consistent with the given macrostate”, we are certainly open to the criticism of vagueness, but not as much as one might expect.

Let’s proceed to take a probability density, , being careful not to specify its exact nature, and define as containing states for which , where in turn we have chosen so that the total probability to find a system in the region is: (10.73) This, in turn, determines a phase space volume: (10.74) and hence: (10.75) It is then possible to show that16: (10.76) Note that the result does not depend on the choice of . Essentially, while changing does impact on , and possibly by a considerable amount, the consequential change in is somewhat less and in any case counterbalanced by the growth in .

So,does measure the phase space volume of “reasonably probable microstates”, and in a way that does not depend on our choice of what we mean by “reasonably probable” in the thermodynamic limit. It is common in the literature to refer to as a phase space volume, when in truth it is the count of states, determined by partitioning the volume into lumps of .

We will adopt this phrasing but underline the word volume to indicate this subtle distinction. Now, let’s consider an experiment performed on a specified system. At time , we measure a set of macroscopic variables , where the collection is sufficient to uniquely define the thermodynamic state of the system.17 Such a selection is not sufficient to pin down which single microstate the system is occupying.

In fact, we have simply specified a region of classical phase space, , in which there is a high probability that the system will be found, consistent with . We will say that this region has volume which in our units of corresponds to . Don’t forget that we have established that we can be vague regarding “high probability”.

If the system is governed by the canonical distribution, then will be the largest phase space volume consistent with . If some other probability distribution is in operation, then its associated volume must fit inside .

“If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”

Sir Arthur Stanley Eddington, Gifford Lectures (1927), The Nature of the Physical World (1928) “The deepest understanding of thermodynamics comes, of course, from understanding the actual machinery underneath.” Richard Feynman, The Feynman Lectures on Physics 39-1 “The Sands of Time were eroded by the River of Constant Change …” Genesis, Firth of Fifth OceanofPDF.com Introduction to Entropy The concept of entropy arises in diverse branches of science, including physics, where it plays a crucial role. However, the nature of entropy as a unifying concept is not widely discussed—it is dealt with in a piecemeal manner within different contexts.

The interpretation of the concept is also subtly different in each case. This book will draw these diverse threads together and present entropy as one of the crucial physical concepts. It will cover a range of different applications of entropy, from the classical theory of thermodynamics, the statistical approach, entropy in quantum theory, information theory and finally, its manifestation in black hole physics.

Each will be presented in a manner suitable for undergraduates and interested laypersons with no previous knowledge. The book will take an overview of these areas and see to what extent the concept of entropy is being treated in the same way in each, and how it differs.

Key Features: Provides an accessible introduction to the exciting topic of entropy, setting out its manifestations in classical thermodynamics, statistical mechanics, and information theory. Covers applications in black holes, quantum theory, and Big Bang cosmology. Jonathan Allday taught physics at a range of schools in the UK. After attending the Liverpool Blue Coat School, he took his first degree in Natural Sciences at Cambridge, then gained a PhD in particle physics in 1989 at Liverpool University.

Shortly after this, he started work on Quarks Leptons and the Big Bang, now published by Taylor & Francis and available in its third edition, which was intended as a rigorous but accessible introduction to these topics.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

Book Information

• Unique ID: fd68fe985ef8c971
• File Extension: .pdf
• File Size: 19,666,215 bytes (18.755 MB)
• Title: –
• Author: Unknown
• ISBN: 9780367638689, 9780367638665, 9781003121053
• Pages: 532
• Language: English (en)

Reading & Word Statistics

• Estimated Reading Time: 619.05 minutes
• Total Words: 123,810
• Total Characters: 748,745
• Average Words per Page: 232.73
• Average Characters per Page: 1407.42

Most Frequent Words

entropy (593), state (523), energy (514), system (378), one (352), quantum (337), gas (305), time (293), number (265), states (260), now (250), two (248), temperature (227), figure (220), also (218), particles (217), first (214), between (214), heat (208), particle (204), theory (199), work (195), chapter (194), information (192), law (189), probability (186), density (183), different (182), function (178), case (175), using (175), form (173), ensemble (172), black (170), systems (170), matrix (168), distribution (164), however (163), second (161), volume (147), cycle (144), constant (140), physics (136), terms (136), classical (135), way (134), molecules (133), need (131), change (130), value (128), equation (128), thermodynamic (126), point (125), hence (124), within (121), possible (119), process (118), result (117), thermodynamics (114), see (111), use (111), it’s (111), von (110), known (107), used (107), space (105), many (104), given (104), pressure (102), microstates (101), mass (101), physical (99), hole (98), another (98), operator (97), along (96), general (95), equilibrium (95), consider (94), wave (93), ideal (92), set (91), thermal (91), radiation (91), statistical (90), much (89), molecule (89), since (87), example (85), neumann (84), important (84), gibbs (84), section (84), partition (84), box (83), measurement (83), boltzmann (82), evolution (81), matter (80), know (79).

PDF Download