# Energies of Mechanical Fractional-Order Elements: Causal Concept and Kernel Effects
## 3 Causality in Dynamic Laws
### Mathematical Models of Dynamic Systems with Respect to Causality  
Following the above-mentioned philosophical concepts, we can now approach mathematical models of physical processes, primarily diffusion models and dispersion relationships. Considering the latter case, when particles move from an initial state to a final stage, we concentrate on the interactions between them that take place in three consequent regimes (states) [11]:  
* _Initial stage_, when the incoming particles are moving towards one another. Their interaction can be neglected, as the distances between them are sufficiently larger than their free paths. * _Intermediate stage_, when the particles are closer together and the distances between them are less than the free parts, resulting in interactions between the particles. * _Final stage_, when the particles leave the contact domain and the distances between them allow them to again be considered as non-interacting. The fundamental problem attracting our attention here is the condition of causation. We formulate several conditions which are related to this concept [11]:  
* _Primitive causality_: the effect cannot precede the cause. In such situations, the cause and effects should be correctly defined. * _Relativistic causality_: no signal can propagate with a velocity greater than the speed of light in the vacuum; this can be considered as a macroscopic causality condition. The state of _primitive causality_ is more basic and general than the state of relativistic causality. The causality principle implies that some functions describing transients in dynamical problems should obey some properties, i.e., to vanish over a range of values of its arguments (see further comments about the causality of the relaxation functions used in memory integrals). Let us consider a physical system with a time-dependent input (cause) \(z(t)\) and corresponding output (effect) \(y(t)\) satisfying the following conditions:  
* **Linearity**: the system obeys the superposition principle in its simple version, implying that the output is a linear functional of the input \[y(t)=\int\limits_{-\infty}^{\infty}g(t,\tau)z(\tau)d\tau\] (33) where \(y(t)\), \(g(t)\), and \(z(t)\) may represent distributions. * **Time-translation invariance**: a system is _time-translation invariant_ if the input is shifted (forward or backward) by some time interval \(\tau\) and \(z(t+\tau)\) corresponds to \(y(t+\tau)\). In this case, the function \(g(t,\tau)\) should depend only on the difference between the arguments, that is, \(g(t,\tau)=g(t-\tau)\), and the linear functional can be expressed as \[y(t)=\int\limits_{-\infty}^{\infty}g(t-\tau)z(\tau)d\tau=g(t)*z(t).\] (34) The relation (34) is a convolution between the _input_ (cause) \(z(t)\) and the _output_ (effect) \(y(t)\); the correlation function (called the _memory_ or _kernel_) allows us to model the time shift, i.e., the output at time \(y(t)\) corresponding to an earlier moment of the input \(z(t-\tau)\). * **Primitive causality condition**: _the input cannot precede the output_; therefore, the input \(z(t)\) vanishes for \(t<T\), meaning that the same is valid for \(y(t)\). Thus, without loss of generality, \(T\) can be assumed to be zero (i.e., the moment when the input is applied). Consequently, we obtain \(g(\tau)=0\) if \(\tau<0\). \[y(t)=\int\limits_{0}^{t}g(t-\tau)z(\tau)d\tau=g(t)*z(t)\] (35)This point is of great importance for the formulation of fractional models directly addressing the concept of initialization, as discussed next. In support of the general formulation of the causal relationship (35), in Appendix A we provide an example of a causal formulation of [PERSON]'s second law in terms of Riemann-Liouville fractional operators, thereby relating the main examples developed in the reminder of this article to the general concept of causality explored and applied here. Moreover, (35) can be considered as a Volterra equation of the first kind used in a specific manner.