A
polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
is a
macromolecule
A macromolecule is a very large molecule important to biophysical processes, such as a protein or nucleic acid. It is composed of thousands of covalently bonded atoms. Many macromolecules are polymers of smaller molecules called monomers. The ...
, composed of many similar or identical repeated subunits. Polymers are common in, but not limited to, organic media. They range from familiar synthetic
plastics
Plastics are a wide range of synthetic polymers, synthetic or semi-synthetic materials that use polymers as a main ingredient. Their Plasticity (physics), plasticity makes it possible for plastics to be Injection moulding, moulded, Extrusion, e ...
to natural biopolymers such as
DNA and
proteins
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...
. Their unique elongated molecular structure produces unique physical properties, including
toughness
In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing.[viscoelasticity
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...]
, and a tendency to form
glasses
Glasses, also known as eyeglasses or spectacles, are vision eyewear, with lenses (clear or tinted) mounted in a frame that holds them in front of a person's eyes, typically utilizing a bridge over the nose and hinged arms (known as temples or ...
and
semicrystalline
Crystallinity refers to the degree of structural order in a solid. In a crystal, the atoms or molecules are arranged in a regular, periodic manner. The degree of crystallinity has a big influence on hardness, density, transparency and diffusion ...
structures. The modern concept of polymers as covalently bonded macromolecular structures was proposed in 1920 by Hermann Staudinger.
One sub-field in the study of polymers is
polymer physics. As a part of
soft matter studies, Polymer physics concerns itself with the study of
mechanical properties
A materials property is an intensive property of a material, i.e., a physical property that does not depend on the amount of the material. These quantitative properties may be used as a metric by which the benefits of one material versus another ca ...
and focuses on the perspective of
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
.
Because polymers are such large molecules, bordering on the macroscopic scale, their physical properties are usually too complicated for solving using deterministic methods. Therefore, statistical approaches are often implemented to yield pertinent results. The main reason for this relative success is that polymers constructed from a large number of
monomer
In chemistry, a monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or three-dimensional network in a process called polymerization.
Classification
Mo ...
s are efficiently described in the
thermodynamic limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
of infinitely many monomers, although in actuality they are obviously finite in size.
Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and dynamics. The path integral approach falls in line with this basic premise and its afforded results are unvaryingly statistical averages. The path integral, when applied to the study of polymers, is essentially a mathematical mechanism to describe, count and statistically weigh all possible spatial configuration a polymer can conform to under well defined
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
and temperature circumstances. Employing path integrals, problems hitherto unsolved were successfully worked out: Excluded volume, entanglement, links and knots to name a few.
[F.W. Wiegel, ]
Introduction to Path-Integral Methods in Physics and Polymer science
' (World Scientific, Philadelphia, 1986). Prominent contributors to the development of the theory include
Nobel laureate
P.G. de Gennes,
Sir Sam Edwards,
M.Doi,
F.W. Wiegel
and
H. Kleinert.
[H. Kleinert, ]
PATH INTEGRALS in Quantum mechanics, Statistics, Polymer Physics, and Financial Markets
' (World Scientific, 2009).
Path integral formulation
Early attempts at path integrals can be traced back to 1918.
A sound mathematical formalism wasn't established until 1921.
This eventually lead
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
to construct a formulation for quantum mechanics, now commonly known as
Feynman Integrals.
In the core of Path integrals lies the concept of
Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differentia ...
. Regular
integrals
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
consist of a limiting process where a sum of functions is taken over a space of the function's variables. In functional integration the sum of functionals is taken over a space of functions. For each function the functional returns a value to add up.
Path integrals should not be confused with
line integrals which are regular integrals with the integration evaluated along a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
in the variable's space.
Not very surprisingly functional integrals often
diverge, therefore to obtain physically meaningful results a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of path integrals is taken.
This article will use the notation adopted by Feynman and
Hibbs, denoting a path integral as:
:
with