Nice question! Energy is a rather narrow band concept, since it comes from Hamiltonian systems that are integrable, so an integrand of motion is energy. Entropy from the other side is restricted to systems that tend to diffuse in disorder rather than in order. The entropy of human civilization always increases, as a counter example. I think that both of them are not a proper characterization, although I cannot present now a robust alternative. As for the question, energy has more applications than entropy.

Dear Demetris,

What is the entropy of human civilization? Which scientific field does deal with it?

The two basic laws which survied untouched from the nineteenth century are the least action principle, based on the symmetries and supported also by the NOETHER'S theorem regarding energy and momenta, and the maximum entropy principle or II principle of thermodynamcs T*dS>=P*dV+dU.  

"the physics operates with notions  of energy and entropy mainly in thermodynamics:"

The phisics talks about Energy everywhere...entropy  in information theory.

I guess these relationships are looking intrerestiung enough to be discussed keeping in mind I. Prigogine's ideas about the self-organization and live systems creation...

Living creatures  basically fight against both the principles...life is ACTION, the law minimises it...life fights against disorder . Living creatures are ordered and need energy at a good level, not degraded (entropy) for their survival.

It would be interesting to integrate both principle in one only...

In thermodynamics whether,  one deals with equilibrium or nonequilibrium (natural) processes  or changes,  These both characteristic functions; namely   the energy and entropy they do have equal merits and importance then any other  characteristic state functions defined by Legendre transformations  to take into account certain  constrains on the thermodynamics system. While Gibbs (p.56) was discussing the criteria of Equilibirum and Stabilityfor a material system, which is isolated from all external influences, he was using both state functions  like a conjugative  two identities, namely:

I)  (del S)U =< 0     or   with equal merit  II)  (del U)S =>0  . He also mentioned that these two theorems are equivalent since it is always possible to increase both the energy and the entropy of the system or to decrease them both to gether.

For me definition of the entropy  by Max Born (1924)  has more practical advantages to deal with nonequilibirum changes: 

dS = dSext  +  dSint   where  dSext  = Del ( Q/ T)  external,   dSİnt => 0 , Where

Internal entropy change is zero for the equilibrium, and positive definite for the natural processes and changes. That definition of the internal entropy doesn’t put any constrain on the state variables, which describe its macrostate. Therefore as Guggenheim commented; the Entropy tells us the direction of changes taking place in the system. The statistical interpretation of entropy using the Boltzmann connection doesn’t bring us too far. Because of that energy and entropy combined together by Legendre transformation to define new function, which is Helmholtz free energy:

F= U -TS where dF= -S dT - p dV + Sumİ ( ChemPotİ dnİ ).

I should remark that whole statistical thermodynamics relies on the Helmholtz free energy because its computation is easier than the Gibbs free energy which involves (stress tensor in deformed bodies ) P as independent variable.

It seems to me to discuss the physical importance of the energy in the presence entropy or visa versa doesn't bring any one any where!

Don't forget one can calculate entropy by counting the number microstates for any types of statistics, but the absolute value of energy.

Actually, internal entropy concept as penalty function is valid as a first order approximation in the distribution functions according to Prigogine calculations. Therefore chaotic systems or the system, which is far from the equilibrium state can't be treated by the irreversible thermodynamics arguments.

HAPPY NEW YEAR ALL

Dear anonymous down-voter,

Please, try to argue! Please, show your brilliant brain sparks!

to @Stefano:

Dear Stefano,

please take into account that when I have been writing 

"...the physics operates with notions of energy and entropy mainly in thermodynamics:" -

I have then meant that physics operates with them MAINLY in thermodynamics JOINTLY! - not separately either with the energy or with the entropy...

Thanks for your attention!

Regards!

Dear Colleagues; each of us has put some light on the concept of energy and entropy according to our own life long experience and institutions. I would recommend reading the paper of Prigogine 'Evolution Criteria, Variational, Properties, and Fluctuations'. Proceedings of Chicago Symposium, May 17-19, 1965. Where he admits that his minimum entropy production principle works only for the linear systems. At the final stage he was exhausted looking for some thing like entropy, which could be minimized but having universal generality. Similarly, Glansdorff and Prigogine spend a lots of time in their book ‘ Structure, stability and fluctuations (1971), and then they ended up with the classical theory of stability, namely:   The second variance lof entropy is negative, and  the time derivative of the second variance of entropy is positive definite. When they needed some help for the justifications,  they asked help from Einstein theory of fluctuations. (SEE: Prigogine ibid:(p.6) as well as Glansdorff and Prigogine (p.96). Actually these two excellent studies showed me that we haven’ done very much progress more then what our grand fathers did almost a century ago.

The rigorous mathematical derivation of second law of thermodynamics was given by Caratheodory who rely on the human experience that ‘’ heat cannot flow from cold body to a hotter one without external control’’ (SEE: Caratheodory's axiom in  PDE by I Sneddon 1957, p.41, which is nothing but irrevesibility principle). 

Anomalous Melting behaviour of    He3 ( 1.0-1.2 K)   as we discussed previously in RG, one has to extract heat  (negative heat of fusion) from solid to melt that means superheated solid is possible or visa versa,  one has to supply heat (enthalpy) to liquid for the solidification  (supercooling state). Best Regards

In our simulation work on the grain boundary grooving under the uniaxial tension, which relied on the theory of irreversible thermodynamics of surfaces and interfaces in deformed solids by Ogurtani, we were able to tract down the time variations of the Global Gibbs free energy of the system. What we have observed at the initial stage of the Groove tip displacement 'the maximum entropy production principle' [the time derivative of second entropy variance is positive]  was obeyed and then process switch to ' the  minimum entropy production ' as postulated by Prigogine [ the time derivative of the second variance of entropy is negative] , finally chaotic stage took place, which ended up with catastrophic failure of the body!(SEE in RG: Phil. Mag. 2011 pp. 1-26)

As I have pointed in that paper, that  this appearant discrepancy is due to fact that Prigogine was dealing with the isolated system. On the hand our realistic system was exposed to isobaric isothermal irreversible process obeying Planck Criteria. 

This was  the first time in the literature that was shown that the both principles are operating consecutively  in the dynamics systems if one tracts down the evolution of system in logarithmic time scale (thermodyanic time) otherwise it might miss the initial stage which seems always obeys' Max rather then Min.

Energy is related to reversibility and/or time symmetry. Therefore energy is also relevant to any system that is interacting with a system that is reversible or has time symmetry. 

Entropy is relevant whenever one want to get information about a complex system.

An important special case is thermodynamics where one wants information about the system in order to extract energy from the system.

Dear Peter,  If you are talking about  the classical or quantum Hamiitonian  mechanics  you may be right. These are  typical concervative system, and their dynamic behavour  are represented by even order time derivates.  That is the reason you have time invariance ( symmetry or ) parity any way you label it ,...reversibility.  No dissipatiom of energy, no dead no birth. But they can't represent evolution in time. I was talking about the energy as defined in thermodyanics but nothing else.  

Similarly I don't like to give  thermodynamic entropy any misterious meanings. I know also its limitations. 

Dear Tarik. Thermodynamics refer to classical mechanics in the sense that energy of a thermodynamic cannot be defined without a reference to the concept of work.

in my opinion, the most fundamental aspects are the concept of complexity for the entropy and generalized force for the energy. For instance, referred to financial markets in Econophysics, I can define its entropy and energy without any other reference to physical variables.

Dear Jose Energy and entropy can sometimes be defined for economic systems. The problem seems to be that in physics the energy is exactly conserved while it is only approximately conserved in economy. Therefore the predictive value of thermodynamics is better than the predictive value of economy.

Dear Peter, you are perfectly right.  I have been  always stressing that point in my lectures. For me  the classical mechanics or the physics at large still the foundation of modern natural sciences besides my good loyal friend  mathematics. Thanks for the comments.

Best regards

Note: I had the luck while I was a graduate student at Stanford in 1960-1964, having  the Late Professors Eric Hutchinson (Student of Guggenheim) and van Ryselberghe ( de Donder was his mentor and coworked 'Affinity'), who were lecturing thermodynamics in any level, and I was also enjoying auditing the statistical mechanics courses of  Paul Flory and Felic Bloch in Chemistry and Physics departments., respectively. 

Mechanics is different from thermodynamics. In some cases, when they come together , better is to take one or another approach ( for example in simulation of plasma confinement ).

Dear Danilo; You are definitely right! if one is a theoretician; he/she shouldn't be shy in dealing with every aspect of the natural sciences including mathematics. At least having enough working knowledge and some intuition to guest what tool she/he needs to solve that problem. Then she/he gets that tool and learns how to use it very effectively. I am 82 years old still learning the new tools if I thing useful for me!! N your case one should definitely to be expert in non-equilibrium thermodynamics as well as on the statistical physics with some knowledge in the stochastic processes. Once I have had to read the article of Chandrasekhar on the random flight theory published in modern physics in early fourties to formulate the vacancy diffusion in crystalline solids. Since I knew the Fourier transformations well enough it was not difficult for me to follow him closely. Then that topic became so popular among those people dealing with the dynamical many body system. Regards

   Without having the deep knowledge in classical mechanics and physics one couldn’t go too far in the quantum mechanics as well as in non-equilibrium thermodynamics of realistic systems. My starting idea in the development of irreversible thermodynamics of surfaces and interfaces was the simple concept of dissipation of mechanical energy: FORCE by VELOCİTY. When it comes to formulate the surfaces and interfaces for deformable solids one should know about the solid mechanics as much as a person specialized in that topic as a mathematical physicist having working knowledge in the large spectrum of applied mathematics and numerical methods. Then one may do some thing original and to be used in practice. BEST REGARDS 

@Tarık Ömer Oğurtanı

Dear Tarif, my respect to you and your  way of thinking and all that! You are strongly right -  yet the problem posted  you  slightly changed - what do you think of the entropy-information stored by a system which ... is out of   interaction to each other? For instance, does it exists and could you find the entropy of a free ideal NON interacting to each other  penetrates each other without changing their Newton dynamics motion?

And then how it would be chnged if the gas under regrad became interacting?

Regards!

Dear Anatolij,  as once Prigogine -after getting exhausted in searching- admitted that it is impossibility   to find any universal quantity such as entropy,  which can be used to maximize or minimize  in order to get the stationary nonequilibrium state of a  system similar to  our naive entropy concept, which fails beyond the first  term of the particle velocity distribution function in the Chapman -Enskog model. Where the zeroth term as you know corresponds to Maxwell-Boltzmann (thermostatic) and the first term justifies the irreversible dissipative systems. From the mathematical point of view,  the exactness  of  the Pfaffien differential equations, which is the foundation of the existence of temperature and entropy twins relies on the simple expression  of X.Curl X=0 that so to speak implies on the our natural observation that the heat goes from the hot region to the cold region!!!   Similarly there are some paradoxes in the irreversible thermodynamic theory as far as the coupled reactions are considered in the molecular level as Prigogine pointed out  in his manuscript (See: p.25 in his book). Actually in my variational treatment of irreversible thermodynamics I have mentioned that ı am not looking for the extremal solutions since they are corresponds to the equilibrium states, all I am looking to get the problem into the variational form properly including the variability of the boundary conditions and the effects of the deformation field. Probably I was the only one tackle this problem in heavily deformed systems, and producing a manageable boundary value problem for two very popular systems such as isochoric and isobaric isothermal OPEN systems.   Best Regards

 I think the philosophical foundation of thermodynamics lies on the frame work of Information Theory or visa versa, which manipulates our ignorance on any system and events in the numerical format to be used for the prediction of the future behaviour relying on our present knowledge of that system. Where Entropy is metric to measure our ignorance!!!

SEE: Non-Equilibrium Thermodynamics Variational Techniques and Stability. Chicago Conference 1965. Prigogine on the Evolution Criteria pp:1-16

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity