System and the phase

The system

We defined two distinct layers. The layer, which contains matter, lay over the fundamental layer. What is important for us is the Intersection of these two layers. Our Picture forms. We know that matter occupies space. Thus, the First thing we must do is to identify precisely the region that matter occupies in the space and specify exactly its boundaries. This must be the first step.
As we specify the boundaries of the region that matter occupies our System is defined. We call this system a Material System.
Of course the system we defined could have a lot of properties but here in our context the first property that we must specify before any other properties, is the region the system occupies in the space and its boundaries. Without these boundaries the system has not yet defined.
Two characteristics of this region are Shape and Volume. After defining the system, the next important property we must specify before specifying other properties is the Distribution of matter in this special region of space.
Thus, we must first specify the region of the distribution of matter ((The geometry of the distribution of matter)). Then we must identify the distribution of matter itself.
The characteristics of the region together with the distribution of matter form the system. From now on, we talk about the system.

The property Distribution within the System

We introduce two distinct group of properties of the system. The qualitative properties and the quantitative properties.
We want to introduce another concept that seems abstract but is useful. We want to extend the concept of distribution to all properties of the system We talk about the distribution of a property. We have Spatial distribution and Temporal distribution of a property within the system.

The appropriate properties of the System

We have already talked about the properties of the system in general. Material systems exhibits a lot of properties. What are other important properties of the system?
Of course it depends on the system and the way we want to study the system. It means, in which aspect we want to study the system. Thus, first we decide in which aspect of the system we are interested, then according to that aspect we choose the appropriate properties . The appropriate properties are those properties that in the best way describe that aspect of the system.
Some important properties that needs more attention are Chemical properties and Physical properties,Uniformity,Pattern,Order.

To complete our discussion about the system and its properties an important thing to notice is that here we have not restricted ourselves to some special properties. We have not defined any limit for properties. We are talking about any property.

The definition of The Phase of the System

We have specified the system and we talked about the properties that a system could possess and we are going to talk about what we call The Phase.
We have discussed that we become interested in some properties depend on the aspect of the system we are going to study, which as much as the system is well-described by these properties as they are more appropriate properties. These properties form a Set.
By choosing the properties we are interested in, the Set of the Properties of The System is well-defined. Until now we only specify these properties, we have not yet assigned a value to them.

The Phase is a collection of determined properties we are interested in. It means we have specified the properties and we determined each of the properties and assigned value to them.
The change in each of these properties results in a change in the phase. So long as we have fixed properties, we have also fixed phase. An infinitesimal change in each member of this collection results in a new phase, what we know as a Phase Transition .

The Study of the System

Here we want to describe in some detail what is happening during the study of a system.
During our study of the system, we consider a lot of aspects of the System. To study each aspect, we choose some appropriate properties. These properties form a Set. The Phase is this collection of determined properties. In general, the system possesses a property and this property while external factors outside the system have not effected this property, remains constant.
we study these properties in two cases : One case when other parameters are fixed, it means observing the spontaneous behaviour of the system. Another case is during changes of an external parameter, in which we see if the system changes or not.
Suppose we want to study the system under the effects of an external parameter. By creating a disturbance at least one of the members of the set of the appropriate properties will experience a change in its value according to this disturbance. Each one of these situations, it means each time we determine the set of all properties, correspond to a phase. In other words, the phase of the system changes. The system undergoes a phase transition. To have a complete description of the behaviour of the system we determine the variation of all the properties of the set of the appropriate properties.In fact, we determine the variation of all these properties due to the disturbance. In fact, we have succeeded to determine all the phases that the system could have. It is also convenient to gather all these phase in one set.
We call this set The Phase space. The phase space is the set of all the possible phases that the system could be in.
We repeat this procedure for other aspects of the system if we were interested in the study of them. It means we repeat this procedure for other sets of properties until we have a complete or better to say satisfactory description of all aspects of the system. After we finish our study of the system, we would have a collection of some collections. We could call it The Set of Phase Spaces. In fact, we summarize the broaden study of the system into a set. The set of phase spaces.

A special case in studying the system

We want to mention one special case in studying the system, in which we do not want to specify the collection of the properties before studying system. After defining the system by specifying the region it occupies in the space and its boundaries, we decide not to define the phase while studying the system. It means we do the same procedure but without specifying any property before we start our study.
We study the system in different situations :
Either we let the system behave naturally and observe the spontaneous evolution of the system or We create a Disturbance while observing the changes in the system.The difference between this case and normal study of the system is that here the phase becomes a Dynamic set rather than a static set.
What does it mean?
It means after we made the disturbance, we observe the behaviour of the system very precisely in order to identify any unusual or unexpected behaviour of the system.Observing the behaviour of the system means be aware of and follow the changes in any element of the system that undergoes a change after disturbance. Unusual or unexpected behaviour also depends on the situation. Sometimes the existence of a discontinuity in one of the parameter of the system is unusual and sometimes the direction of the changes of a parameter and sometimes the change itself is regarded unusual or unexpected.
The phase with a dynamic set means that each time we notice a change in a property that somehow could describe some aspects of the system, we add it to the collection of the properties that we are interested in. Thus, the elements that form the phase could be different during the study of the system. This is what we call phase with a dynamic set because we have not decided which property we are interested in before studying the system and it is in contrast with the phase with a static set which before the study of the system we have determined the collection of properties which we are interested in.
One of the results of this way of study of the system is that we put ourselves in the situations, in which we could observe unexpected or maybe strange behaviour of the system.One of the results of this way of study of the system is that we put ourselves in the situations, in which we could discover something new or we could discover some hidden properties of the system that we could not notice them in normal situations .
Of course by adding that hidden or newly discovered property to the collection of the properties that we are interested in, we have new phases of the system that have not existed before. And we could have new type of phase transitions that reveals new aspects of the behaviour of the system that we did not know it before.

Topological phases of matter

In order to give an example of this special case we want to talk about topological phases of matter which was a discovery based on the works of some researchers in recent decades, so that is a new discovery and is useful in understanding the behaviour of matter.
Supposing that we are familiar with topology, we do not need to explain something new we only need to follow the procedure we mentioned in previous sections in order to study a system in general. According to normal study of the system, we first define the system and then we choose a set of properties, which we called appropriate properties. The key point is exactly here that distinguished different type of the study of the system which usually there exist different names for them like thermal study or mechanical study of the system. Moreover, we know that the only difference between these types of study is the different collection of properties. So as topological study of a system .
It is like other types of study. The only difference of it with other types of study of the system, is in the choice of properties that we choose to study. If we choose properties that by using topology as a mathematical tool we could study the system precisely, we say we study the system topologically. We also call the phase of the system a topological phase because in the set of appropriate properties there exist at least a property which we use topology to study that property. We could call them Topological properties.
And we also have a special kind of phase transition we call topological phase transition.