Swami Vivekananda has written extensively about his ideas of caste in his Complete Works.
But has he written anything explicitly on the psychological theory of Varna?
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Here is an example of one such writing. Just a note of caution. Vivekananda uses the word caste both in the sense of Jati and Varna. It is clear from the context what he means. I have added in square brackets the word Varna or caste in appropriate places to help the reader.
I am adding a brief summary of the Vivekananda passage as requested in the comment.The essence of the passage is that variety is the sign of life. Men of different mental capacity or varna should be allowed to freely express their various natures for the good of society.This is the true idea of varna. The degraded Varna practised by Hindus prevents this flowering of various minds by segregating persons of different mental capacities into groups and then stopping themselves from freely expressing themselves. The idea is that we should allow people to contribute to the betterment of society in various ways regardless of their birth in a specific group. Europe has freely allowed men to becme poets, scientists, priests etc regardless of their origin while India has closed all paths to contribute to the progress of society for a large fraction of its people. No one can tell how many genuises of Sudra caste have been lost due to such restrictions. It is not as if people of other castes did not lose out. Not every person of Brahmin caste had minds of Brahmin varna. Yet most Brahmins were forced to priesthood and remained extremely poor. One can say the same thing about Kshatriyas and Vaishyas. Moreover the degraded varna system invented a category of people who were considered to be outcastes which implied that they were not part of the purusha of the Rig Veda. This is of course absurd since Brahman is all. These people suffered extreme deprivation and India lost out their creative contribution to society. The net result of this defective interpretation is that India has been the loser.
A PLAN OF WORK FOR INDIA (Written to Justice Sir Subrahmanya Iyer from Chicago, 3rd Jan., 1895.)
It is with a heart full of love, gratitude, and trust that I take up my pen to write to you. Let me tell you first, that you are one of the few men that I have met in my life who are thorough in their convictions. You have a whole-souled possession of a wonderful combination of feeling and knowledge, and withal a practical ability to bring ideas into realised forms. Above all, you are sincere, and as such I confide to you some of my ideas.
The work has begun well in India, and it should not only be kept up, but pushed on with the greatest vigour. Now or never is the time. After taking a far and wide view of things, my mind has now been concentrated on the following plan. First, it would be well to open a Theological College in Madras, and then gradually extend its scope, to give a thorough education to young men in the Vedas and the different Bhâshyas and philosophies, including a knowledge of the other religions of the world. At the same time a paper in English and the vernacular should be started as an organ of the College.
This is the first step to be taken, and huge things grow out of small undertakings. Madras just now is following the golden mean by appreciating both the ancient and modern phases of life.
I fully agree with the educated classes in India that a thorough overhauling of society is necessary. But how to do it? The destructive plans of reformers have failed. My plan is this. We have not done badly in the past, certainly not. Our society is not bad but good, only I want it to be better still. Not from error to truth, nor from bad to good, but from truth to higher truth, from good to better, best. I tell my countrymen that so far they have done well — now is the time to do better.
Now, take the case of caste — in Sanskrit, Jâti, i.e. species. Now, this is the first idea of creation. Variation (Vichitratâ), that is to say Jati, means creation. "I am One, I become many" (various Vedas). Unity is before creation, diversity is creation. Now if this diversity stops, creation will be destroyed. So long as any species is vigorous and active, it must throw out varieties. When it ceases or is stopped from breeding varieties, it dies. Now the original idea of Jati [i.e. Varna] was this freedom of the individual to express his nature, his Prakriti, his Jati [Varna], his caste [Varna]; and so it remained for thousands of years. Not even in the latest books is inter-dining prohibited; nor in any of the older books is inter-marriage forbidden. Then what was the cause of India's downfall? — the giving up of this idea of caste [Varna]. As Gitâ says, with the extinction of caste [Varna] the world will be destroyed. Now does it seem true that with the stoppage of these variations the world will be destroyed? The present caste is not the real Jati [Varna], but a hindrance to its progress. It really has prevented the free action of Jati, i.e. caste [Varna] or variation. Any crystallized custom or privilege or hereditary class in any shape really prevents caste (Jati [Varna]) from having its full sway; and whenever any nation ceases to produce this immense variety, it must die. Therefore what I have to tell you, my countrymen, is this, that India fell because you prevented and abolished caste [Varna]. Every frozen aristocracy or privileged class is a blow to caste [Varna] and is not-caste [caste]. Let Jati [Varna] have its sway; break down every barrier in the way of caste [Varna], and we shall rise. Now look at Europe. When it succeeded in giving free scope to caste [Varna] and took away most of the barriers that stood in the way of individuals, each developing his caste[Varna] — Europe rose. In America, there is the best scope for caste (real Jati [Varna]) to develop, and so the people are great. Every Hindu knows that astrologers try to fix the caste [Varna] of every boy or girl as soon as he or she is born. That is the real caste — the individuality, and Jyotisha (astrology) recognises that. And we can only rise by giving it full sway again. This variety does not mean inequality, nor any special privilege.
This is my method — to show the Hindus that they have to give up nothing, but only to move on in the line laid down by the sages and shake off their inertia, the result of centuries of servitude. Of course, we had to stop advancing during the Mohammedan tyranny, for then it was not a question of progress but of life and death. Now that that pressure has gone, we must move forward, not on the lines of destruction directed by renegades and missionaries, but along our own line, our own road. Everything is hideous because the building is unfinished. We had to stop building during centuries of oppression. Now finish the building and everything will look beautiful in its own place. This is all my plan. I am thoroughly convinced of this. Each nation has a main current in life; in India it is religion. Make it strong and the waters on either side must move along with it. This is one phase of my line of thought. In time, I hope to bring them all out, but at present I find I have a mission in this country also. Moreover, I expect help in this country and from here alone. But up to date I could not do anything except spreading my ideas. Now I want that a similar attempt be made in India.
I do not know when I shall go over to India. I obey the leading of the Lord. I am in His hands.
"In this world in search of wealth, Thou art, O Lord, the greatest jewel I have found. I sacrifice myself unto Thee."
"In search of some one to love, Thou art the One Beloved I have found. I sacrifice myself unto Thee." (Yajurveda Samhitâ).
May the Lord bless you for ever and ever!
The Complete Works of Swami Vivekananda, Volume 4, Writings Prose, A Plan of Work for India