Thermodynamics as formulated by Max Born [Z. Physik, 22, 218, 249,282 (1921)] is a mathematical science, which depends on the few axioms {first and second laws   that were put on sound mathematical format by Caratheodory (Math. Ann. 67, 355 (1909) which was also cited above reference}  and the rest is well known mathematical procedures. I would recommend to you the first chapter of book entitled Thermodynamics By Guggenheim, which gives you basics philosophy of Thermodynamics as he calls Thermostatic. More the mathematics I recommend the book entitled 'Elements of Partial Differential Equations by Ian N. Sneddon'' where the sections on the Pfaffian Differential equations and Theory of Caratheodory especially the one on the Second Law is very important. Because the Second Law as it is today relies on our experience in life that certain changes of state are not physically realizable: ''Heat cannot flow from a cold body to a hotter one without external control.'' Caratheodory's axiom is:

Arbitrarily near to any prescribed initial state there exist states which cannot be reached from the initial state as a results of adiabatic processes.

There is a mathematical theorem related to the Pfaffian Differential Form (PDF) 

Delta X=  SUM  Xk  dxk   which tells that for a given point Go there exit points G which are  inaccessible from Go  along curves  for which  Del X=0 , the corresponding PD equation  Del=0  is integrable. That follows from this theorem that  there exist functions   Mhu(x )   and   S( x )  with the property

Mhu(x) Del(x)  =  dS(x)  !!!!!!!!  Where   S(x)  is called  ENTROPY ,

Mhu is, apart from a multiplicative constant, a function only of the empirical temperature of the system, which is written as  1/T.

 By using absolute scale for the termperature  in the appropriate manner,  above equation may have into the familiar form:   Del Q /T  =  dS

THE CONDITION OF INTEGRABILITY IS    X.  Curl  X  =0

You must understand that dynamics of a single particle (or a few particles) is a different problem than thermodynamics which treats macroscopic amount of material consisting of 10^23 or more microscopic particles. Classical thermodynamics of course does not even consider these particles at all but the statistical mechanics does. So thermodynamics is not a microscopic theory.

Dear Chatterji,

I know this is the popular view about thermodynamics, but, there is a certain relation between a macro physical quantity and its micro quality, which involves the relation between thermodynamics and dynamics.

The fundamental postulate of statistical mechanics includes the equal a priori probability postulate, this postulate does not need to be considered in thermodynamics, so the valid range of statistical mechanics cannot cover thermodynamics.

Dear Tarik,

For Carnot cycle, how to explain∮δW/T=0?

Dear Tang,  δW  is not an exact differential  that means it can't be derived from a gradient of a scalar function as such  dr. grad V,   which results zero for  any closed line integral for  any given domain of interest.  Above integral is also called path dependent.  The value of above integral should corresponds to the energy dissipation or generation 

Dear Colleagues  Tapan and Tang,     If you are a mathematical physicist you should be the master of both disciplines that means you have to be aware of not only their superiorities but also their limitations when it comes to apply them for specific problems in practice as one faces in engineering as well as in natural sciences.  Look at  the statistical thermodynamics  very carefully [Fowler and Guggenheim] then you will realize that you can only treat the isochoric systems. That means we have to deal with those systems, which are mechanically isolated from their surroundings as such that they are not exposed to any surface tractions and body forces, that can be represented by Helmholtz free energy, F involving only the linear combination of energy, E  and entropy, S   as state functions.  On the other hand Statistical mechanics has great computational power which can't even be dreamed  of  in the field of Thermodynamics whether it deals with equilibrium or non-equilibrium systems. For example, the  statistical mechanics calculations by Prigogine on the Chapman-Enskog model of non-uniform gases  set the limit of validity of irreversible thermodynamics by showing that in the expansion of the particle velocity distribution function zeroth term  fo leads to thermostatics (no entropy production),  the use of  f o +  f1  gives exactly the same results of irreversible thermodynamics relying on Gibbs' relation.  But including the second order term  fo  + f1  +  f2  different results are obtained for thermodynamics and kinetics theory ( a result of the H-theorem).  BEST REGARDS

As I have shown rigorously  in  our  two recent papers we can handle  the  isochoric and isobaric system respectively under the internal external and external stresses  by using variational formulation of  irreversible thermodynamics of deformable bodies. We have also presented its  recent  applications on the surface morphological revolutions in thin solid films, formation of quantum dots, catastrophic failure of microelectronic devices through electromigration and/or grain boundary growing   enhanced by uniaxial stresses, the healing affects of surface cracks and undulations by applied electrostatic force driven surface diffusion etc. Non of these problems ever have been treated or even attempted to be deal with by statistical mechanics or thermodynamics. Most of the systems we were dealing with were  thin films having nano metric dimension where the  wetting layers (QD) having thicknesses in the range of  the fraction of a nanometer. BUT all these don't mean that the statistical mechanics should be overlooked as a computational tool, which has limited applications in technological problems that one faces in the daily life.