Single-molecule thermodynamics

In thermodynamics, single-molecule thermodynamics is the study of the thermodynamics of an individual molecule or single molecule in a system or of the measurement and description of quantities, variables, and or functions at the single-molecule level. The interaction of a single photon causing a single retinal molecule to do movement work, i.e. straighten or bend in conformation, might be a descent model example of a single molecule thermodynamics case study.

Extensivity issue
The subject of thermodynamics of a single molecule or for that matter single particle (single particle thermodynamics) is a tricky subject, to say the least. The science of thermodynamics, regardless, is applicable in its governing nature to both the single particle and the single molecule. This is captured well in subject initiator French physicist Sadi Carnot's 1824 pronouncement that: [5]

“It is necessary to establish principles [of thermodynamics] not only applicable to steam engines but to all imaginable heat engines, whatever the working substance and whatever the method by which it is operated.”

In this framework, the single molecule or single particle is defined as the "working substance" whose properties are governed by energy conservation (the mechanical equivalent of heat is a fixed ratio, whatever the means of transformation of work in to heat or heat into work) and entropy increase (caloric is not a reestablishable indestructible quantity).

This universality proclamation, by Carnot, that thermodynamics applies to "whatever the working substance", as rooted historically in the universal framework of Boerhaave's law (1720), and prior to that experimental testing on various substances with the Papin digester (1680s), was eventually stylized in the mathematical language of German physicist Rudolf Clausius (1850-1865), according to which certain variables were defined as extensive (extensity), path-independent, state variables, or state functions; whereas others were defined as intensive (intensity), path-dependent, variables; all of which seems to be crouched in the elusive "condition for an exact differential" framed around the internal atomic interpretation of an Avogadro number sized system (or working substance). Hence, when when theorizing outside of the normal thermodynamics black box, in subjects where particle or molecule count approaches a countable number, hesitancy comes to the fore in regards to the means, usage, quantification, and applicability of the standard model version of the extensive variables and intensive variables. This is evidenced well in Mexican thermodynamicist Karo Michaelian's 2005 justification for the use of Prigogine entropy in the thermodynamics quantification of species of ecosystems:

“The total time change of entropy of the ecosystem (as for any open system) is a sum of an external term of no definite sign, and in internal production term of positive definite sign as required by the second law of thermodynamics:

 \frac{dS}{dt} = \frac{d_e S}{dt} + \frac{d_i S}{dt} \,

 \frac{d_i S}{dt} > 0 \,

In the spirit of the virial expansion for a thermodynamic system communicating through n-body interactions, the total change of entropy of the ecosystem can be written as a many-body expansion of entropy changes due to interactions among individuals and among individuals and their external environment. Such a many-body expansion is obviously in complete accord with the extensity property of entropy.”

The term “virial expansion” seems to have been introduced in 1901 by Dutch physicist Heike Kamerlingh-Onnes. [7] In any event, Michaelian's statement that it's "obvious" that the extensity property of entropy (in the Prigogine-interpretation he uses) carries over absolutely into the quantification of n-bodies interacting in an ecosystem.

What if, however, n = 10, as was the case during the great chimpanzee war (1970-1974), studied and followed by Jane Goodall in the Gombe national reserve, Nigeria, where following a series of growing tensions, a splinter group of seven males and three females with their young split off from the main group and began to form their own troop; after which a war began, whereby subsequently the boundary entropy changes  d_e S \, involved entropy quantifications such as the heat of battle, at the front, territory-infringement murders, etc., as well as the internal entropy changes  d_i S \, internal work of total species genocide. [8] In this simple example, it is not at all "obvious" how the condition for an exact differential, which mathematically defines the extensity property of entropy, translates over to these types of heat measurement quantifications.

It sum, it is generally understood that measurements made on large ensembles of molecules are routinely interpreted using thermodynamics, but that the normal procedure of thermodynamic application diverge to a certain extent when measurements are made on single molecules. [1] This is not to say that the laws of thermodynamics are violated, but rather the assumptions behind the various state variables, measurements that quantify these variables, and state functions or equations that describe the behavior of the “system”, “working body”, or “working substance”, which in this case would be one molecule, require reformulation and a regression to first principles.

Thermodynamics as a scientific discipline is phenomenological, and as such is not dependent on concepts relating to particle interactions, such as is the case with statistical mechanics, but rather the particle interactions are dependent on the governing nature of the thermodynamics of the system. This is often overlooked by many scientists who falsely assume that statistical mechanics delivers the needed conceptual underpinnings. A well-formulated reminder of this fact can be found in a statement by American ecological engineer Robert Ulanowicz: [3]

“Thermodynamics is a self-consistent body of empirical knowledge that may be readily verified without any recourse to microscopic particles. The aim of statistical mechanics was to lend credence to the atomic hypothesis by demonstrating that it could be related to the more fundamental empirical laws. A failure of this effort would have meant trouble for the microscopic hypothesis, not for thermodynamics.”

One trend that one will find with small systems (1-1000 particle range), according to recent nanothermodynamics computer simulations, is that entropy becomes nonextensive, which raises possible issues on the integrating factor of the inexact heat differential, among other possible theoretical issues. In the 2005 article “Nonextensivity and Nonintensivity in Nanosystems” by Ali Mansoori and Pirooz Mohazzabri, they summarize that systems containing particles (argon-like particles) in the 2-1000 range show deviation from standard macrothermodynamics: [1]

“Using extensive molecular dynamics simulations, we have investigated the extensivity of the internal energy and entropy as well as the intensivity of temperature and pressure in small thermodynamic systems. Atomic systems consisting of n³ (n = 2, 3, …10) argon-like particles interacting through the Lennard-Jones potential energy function have been studied. It is found that in small systems, contrary to macroscopic systems, internal energy and entropy are nonextensive whereas temperature and pressure are nonintensive. These deviations from macroscopic thermodynamics, that continue to remain detectable even in systems containing as many as 1000 particles, are in agreement with theoretical predictions.”

By extrapolation downward, one may expect to find further formulaic derivation when the system consists of one atom, one molecule, or one particle (single particle thermodynamics).

Human thermodynamics
In the various subjects of human thermodynamics, analysis is generally done on systems consisting of an integer number of reactive human molecules, generally greater than two or more, upwards to the size of the population of one country, or boundaried systems consisting of upwards of one billion human molecules, in the case of India and China. [2]

An example of a single molecule thermodynamics subject in human thermodynamics is psychodynamics, in which thermodynamic formulation and description of the states of mind of an individual human molecule are attempted in order to develop theory and give psychological guidance and counseling, a field largely developed by Austrian psychologist Sigmund Freud as outlined in his 1895 "A Project for Scientific Psychology".

See also
Microthermodynamics
● Macrothermodynamics
Phenomenological thermodynamics
● Small-systems thermodynamics
Molecular thermodynamics
Nanothermodynamics

References
1. Keller, David, Swigon, David, and Bustamantez, Carlos. (2003). “Relating Single-Molecule Measurements to Thermodynamics” (abs), Biophysics Journal, 84(2:1): 733-8.
2. List of countries by population – Wikipedia.
3. Ulanowicz, Robert E. (2000). Growth and Development - Ecosystems Phenomenology (Ch. 2: The Perspective, pg. 11). New York: toExcel Press.
4. Mohazzabri, Pirooz, and Mansoori, Ali G. (2005). “Nonextensivity and Nonintensivity in Nanosystems: A Molecular Dynamics Simulation” (abs), Journal of Computational and Theoretical Nanoscience, 2(1): 138-47(10).
5. Carnot, Sadi. (1824). Reflections on the Motive Power of Fire: and on Machines Fitted to Develop that Power. Paris: Chez Bachelier, Libraire, Quai Des Augustins, No. 55.
6. Michaelian, Karo. (2005). “Thermodynamic Stability of Ecosystems” (abs), Journal of Theoretical Biology, 237(3): 323-35.
7. Virial expansion – Wikipedia.
8. Thims, Libb. (2007). Human Chemistry (Volume One) (chimpanzee war, pgs. 61-62). Morrisville, NC: LuLu.

Further reading
● Rubi, J.M., Bedeaux, D. and Kjelstrup, S. (2006). “Thermodynamics for Single-molecule Stretching Experiments” (abs). Arxiv.org.
● Rubi, J.M., Bedeaux, D., and Kjelstrup, S. (2007). “Unifying Thermodynamic and Kinetic Descriptions of Single-Molecule Processes: RNA Unfolding under Tension” (abs), J. Phys. Chem. B. 111(32): 9598-9602.
● Fiore, Julie L, Kraemer, Benedikt, Koberling, Felix, Edmann, Rainer, and Nisbitt, David J. (2009). “Enthalpy-Driven RNA Folding: Single-Molecule Thermodynamics of Tetraloop-Receptor Tertiary Interaction.” (abs), Biochemistry, 48(11): 2550-2558.
● Chatterjee, Debarati and Cherayil, Binny J. (2011). “Single-molecule thermodynamics: the Heat Distribution Function of a Charged Particle in a Static Magnetic Field” (abs), Journal of Statistical Mechanics: Theory and Experiment, Vol. 11, March.

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