Mechanics can be founded on a principle relating the uncertainty delta-q in the trajectory of an observable particle to its motion relative to the observer. From this principle, p.delta-q=const., p being the q-conjugated momentum, mechanical laws are derived and the meaning of the Lagrangian and Hamiltonian functions are discussed. The connection between the presented principle and Hamilton's Least Action Principle is examined. Wave mechanics and Schrodinger equation appear without additional assumptions by choosing the representation for delta-q in the case the motion is not trajectory describable. The Cramer-Rao inequality serves that purpose. For a particle hidden from direct observation, the position uncertainty determined by the enclosing boundaries leads to thermodynamics in a straightforward extension of the presented formalism. The introduction of uncertainty in classical mechanics formulation enables the translation of mechanical laws into the wide ranging conceptual framework of information theory. The boundaries between classical mechanics, thermodynamics and quantum mechanics are defined in terms of informational changes associated with the system evolution. As a direct application of the proposed formulation upper bounds for the rate of information transfer are derived.
Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter
θ and its corresponding extensive quantum operator
O^ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter
θ is quantified by its quantum Fisher information
Fθ. The framework is based on an exact integral representation that relates
Fθ to the autocorrelation function of operator
O^. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation:
ΔθΔO≥kBT with
ΔO≡∂θ⟨O^⟩Δθ and
T is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.
Thermodynamics is the science that deals with all the phenomena that involve the transfer of energy, i.e. heat and work. Its development started in 1824 with the efforts of Sadi Carnot to improve the closed-cycle steam engine discovered by James Watt in 1764. In 1850, R. Clausius laid the foundations of the laws of thermodynamics. Soon thereafter a conflict between the second law of Thermodynamics and the laws of mechanics was pointed out by Maxwell in 1871 and was illustrated clearly by what has come to be known as Maxwell’s demon. This conflict was addressed by Brillouin (1949) based on the work of Szilard (1929) and of many others. They all claimed that although the objective state of a system is fully describable by mechanics, classical or quantum, and evolves according to the laws of mechanics, some states of the system as perceived by an observer are subjective and reflect the lack of information the observer has about the actual mechanical state of the system. This point of view is, in effect, the information-theory interpretation of thermodynamics which, currently, is widely accepted. The central point of this paper is to describe the reasons why the information-theory interpretation of thermodynamics is contrary to physical reality. It shows that a logically viable hypothesis which reconciles mechanics with thermodynamics is the existence in nature of physical states that have objective uncertainties broader than those implied in quantum theory as it is traditionally formulated. The consequences of this hypothesis are presented in the Unified Quantum Theory of Mechanics and Thermodynamics, by Hatsopoulos and Gyftopoulos.An initial version of this paper was published inJuly of 2006 in the proceedings of ECOS’06, AghiaPelagia, Crete, Greece.