On Some Aspects of Systems in Thermofluiddynamics
MICHAEL LAUSTER
Faculty for Economics
University of the German Armed Forces Munich
Werner-Heisenberg-Weg 39, 85577 Neubiberg
GERMANY
Abstract: This paper deals with the mathematical foundations of systems in thermofluiddynamics. The basic ideas for the formulation of a consistent theory with a minimum of primitive elements are given. From a single fundamental relation any information on the physical object in regard is retrieved by differentiation processes. It is shown how and under what conditions the results of classical mechanics fit into the theory. Furthermore, restrictions and defects of traditional formulations are shown. The advantages of a system theoretical approach to thermofluiddynamics are explained.
Key-Words: system theory; thermodynamics; fluiddynamics; space; time; Gibbs space; energy; non-equilibrium; dissipation
1 Space, Time, and Generic Physical Quantities
One of the major tasks of sciences is the compilation of data derived from empirical experiences, to store them in a standardized way, and to share them amongst scientists.
In the natural sciences, mathematics has proven to be the appropriate choice for this task: a sharply defined vocabulary and a singularly strong grammar make it a unique language which is spoken worldwide.[1]
Referring to physics and the engineering sciences two options to formulate empirical phenomena have arisen.
Following the evolution-generated experience of reality physical objects are placed in a space with length, breadth, and height.[2] This space is homogeneous and isotropic in each of its directions.[3] Objects move through the three-dimensional space and their movement is characterized by a time co-ordinate flowing uniformly from past to future.[4]
The appropriate mathematical model of such a space is a three-dimensional Euclidian space with orthogonal axis and equal scales in each direction. The origin as well as the angular position of the axis may be set arbitrarily.
As for the time co-ordinate the same holds: origin and unit length of the scale may be chosen arbitrarily. A space comprised of three spatial and one temporal co-ordinate will be called a parameter space.[5]
In the parameter space the path of objects taken during their movement is depicted by a continuous curved line called trajectory. Provided a (positive) real number for the mass is attached to the trajectory, the mathematical model for a body is given.[6]
With the Hamiltonian theory, classical mechanics of multi-body systems found its final formulation. Here, a set of 2s-many mutually independent variables xi and pi, i = 1,2,...s, called the canonical variables position and momentum for each of the s-many particles in regard are used to describe the dynamics by a set of 2s-many coupled first order partial differential equations of the form
1)
Equations are named the canonical equations of motion.
The canonical equations combine both options for the description of empirical phenomena. While the partial derivatives belong to the phase space, the total ones stem from the parameter space.
The Hamilton function H represents the total energy of the particles and t is the time parameter. The evolution in time of any property of the particles represented by a variable F may be expressed with the aid of the Hamiltonian H:
2)
The expression on the right side of equation is identified with the famous Poisson bracket by definition
3)
Substituting the property F by H itself then with
4)
a conservation law results stating that the total energy is a conserved quantity. Condition shows a strong connection between the time t and the total energy given by H.[7]
Regarding the definitions from mechanics
5)
it is obvious that any information on the multi-body system, i.e. the velocities of the particles as well as the interacting forces may be derived from the system describing function H by differentiation.
Taking a close look at the Hamilton mechanism, the canonical equations of motion may be distinguished into two kinds of relations: 1 is a mere definition for the kinematic velocity while 2 is a theorem stating how the momentum of a moving body may be changed. Within the context of Hamiltonian theory, the definition as well as the theorem hold unconditionally.
The second option for the construction of descriptions in the natural sciences came up when classical mechanics was given its final structure and has been intensely used when thermodynamics appeared as a new discipline in physics.[8] It is far more abstract than the first one and does not directly refer to our sensual experiences.
Physical objects are examined with regard to the quantities they exchange with other objects of the same or different kind. The quantities exchanged may be either matter-like, e.g. like substances or immaterial like e.g. linear or angular momentum. As a rule, these quantities should be chosen in such a way that they do not only belong to one specific discipline but are constitutive for the phenomena throughout physics. Quantities fulfilling this condition like, e.g. the total energy, are called generic physical quantities or simply generics.
Every generic physical quantity is mathematically represented by a variable belonging to the set of functions that may be differentiated at least twice. The two-fold differentiability allows the construction of stability criteria.
It goes without saying that only a finite number of generics may be regarded. The creator of the description has to decide which of the effects are important for the purpose in regard and which are not. Once these generics are identified and the respective variables are defined, the description is complete by definition. Depending on the resulting number of generics an abstract space spanned by these variables is generated which is not accessible by our common empirical experience. It is called phase space or Gibbs space. Spatial co-ordinates and the time forming the parameter space do not belong to the set of variables of the Gibbs space.
Each variable may attain values which are represented by real numbers. For macroscopic systems the use of a continuum for the values of a variable is required by hypothesis. Thus, a continuous change of the values of a variable may be established assuring the applicability of the calculus.
Setting all variables of a Gibbs space to a respective value, a single point of this space is referenced. The vector of values is called a state. The junction (in the sense of set theory) of all possible states for an object is called a system.
In mathematical terms a system is a relation of all the variables , of the Gibbs space:
. 6)
is called Gibbs Fundamental Relation (GFR).
The set of variables (as well as the variables themselves) leading to a relation homogeneous of degree one is called extensive.[9]
The differentiability assures that the first partial derivatives of a specific extensive variable with respect to another may be calculated. This delivers the so-called conjugate intensive variables. The notion “intensive” here refers to such quantities that are not additive, i.e. they do not change their values if two identical systems are combined to a single new one.[10]
Objects change their state by exchanging quantities with other objects. Such a change of state is called a process. As a hypothesis, no discontinuous processes occur for macroscopic objects. Therefore, the mathematical picture of a process is a continuous sub-set of the system forming a curved (one-dimensional) line. Each point of that process path, i.e. each state attained during the process, may be assigned a certain value of a time parameter. In other words: At any given time, the object attains a certain state; the inverse is not true. Thus, the process in Gibbs space is the analogue of the trajectory in parameter space.[11]
Both options for the description of empirical phenomena have to be combined to achieve a picture as complete as possible: while in Gibbs space the question is answered how processes run, in parameter space it is expressed where and when the processes take place.
2 Systems in Thermofluiddynamics
In physics, thermodynamics is a prominent example for the application of systems theory. Here a holistic view of the respective phenomena and the acting objects leads to mathematical descriptions that differ essentially from those used in other disciplines of physics. One of the promising new theories in this branch is the so-called Alternative Theory of non-equilibrium processes (AT) created by D. Straub [6]. Its primary purpose is to tackle the problems of irreversible phenomena in thermofluiddynamics. However, in the meanwhile it has been extended to microscopic as well as to electromagnetic systems (cf. [6] and [7]). Its mathematical core has been applied to other quantitative branches of sciences, e.g. national economics.
The mathematical basis of the AT is the formalism first found by Gibbs for thermostatic phenomena. Here every information on the system in regard may be obtained from a single function by differentiation. Falk extended this concept by the insight that linear and angular momentum have to be included as members of the set of variables. This changed Gibbs’ thermostatics to a real dynamics.
Additionally, it is supplemented by the important conclusion of Straub that despite the usual procedures of thermostatics and classical thermodynamics the fundamental set of variables has to be extensive (cf. [5]).
Four primitive, i.e. not reducible elements - variable, value, state, and system – in the meaning described above constitute Gibbs-Falkian dynamics. The only hypothesis necessary to use this method is the possibility to define extensive variables mapping the generics for the respective object.
It is the first step and the major task of the user to identify those variables that are important for the intended description and to neglect those that are of minor interest.
Once all necessary variables are identified, everything works according to the recipe given by Gibbs and Falk. A Gibbs Fundamental Relation exists from which any information is to be retrieved by differentiation.
To get more specific, we look at an object whose generics are - besides its total energy E* - linear and angular momentum, represented by the variables P and L, body forces and momenta F and M, as well as the traditional thermodynamic variable entropy S, volume V, and particle number N. Then the Gibbs Fundamental Relation reads .
The Gibbs Fundamental Relation may be solved with respect to any of the variables. Usually, the total energy E* is chosen to be the dependent variable. Thus, the so-called Gibbs function results:
. 7)
The total derivative of equation , the Gibbs main equation for the system, delivers the intensive variables as the first partial derivatives of the total energy E* with respect to the independent variables:
8)[12]
This defines the linear and angular velocity, the position vector, the angular position, the thermodynamic temperature and pressure as well as the chemical potential, respectively.
The extensivity of the set of variables leads to a GFR that is homogeneous of degree one. For objects from physics the respective factor of homogeneity is the number of particles N of the object. As a rule for all relevant processes in thermofluiddynamics changes below the elementary-particle level do not occur. This results in the constancy of the number of baryons and leptons and therefore in the constancy of the mass m of the object. Hence, for certain applications m may be used as the factor of homogeneity instead of N.
For the Gibbs function being homogeneous of degree one, Euler’s rule for homogeneous functions holds and the total energy E* may be expressed explicitly:
9)
Calculating the total differential of equation and comparing it to , an important conclusion may be drawn: from the product rule of the calculus we find that
10)
has to hold so that and are true without contradiction.
Equation is called Gibbs-Duhem relation. It states that there is a relation between the intensive variables of the system. In other words: linear and angular velocity do not only depend on linear and angular position but are influenced by the thermodynamic variables temperature, pressure, and chemical potential as well. Thus, the effects of motion and thermodynamics may not be separated in general.
3 Matter and Forces
One of the most important facts in GFD is Falk’s observation that besides the thermodynamic variables like entropy, volume, and number of particles, the linear and angular momentum are generic members in the variable list of the GFR. This turns the method from Gibbs’ thermostatics formulated in [5] to a real dynamical theory that includes mechanics with all its kinematic aspects, thermodynamics, and electrodynamics for real matter.
The question is still to answer how classical physics in the form of Hamilton’s mechanics and Gibbs-Falkian dynamics, especially in the formulation of the AT, can be made compatible.
The first step will be a two-fold Legendre transformation that changes the set of variables of the total energy E*. In the variable list of the Hamilton function H the generalized momentum and spatial-co-ordinates appear while forces and momenta are mere definitions within the theory. Therefore, we transform E* according to
11)
and exchange forces and momenta against spatial and angular co-ordinates.
The energy-like variable E is a residual quantity that results, when the total energy E* is reduced by the energy forms of motion. Simultaneously, an aspect of the parameter space is brought into the system description via the variables r and .
The Gibbs main equation then transforms into
12)[13]
The second step is to substitute the increments by total time derivatives and to set the following conditions: